%%% From: Vladimir Chernousov <chernous>
%%% Date: Mon, 15 Nov 1999 16:29:19 +0100 (MET)
%%% To: rehmann
%%% 
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\title{\bf Algebras of exponent 2 over an elliptic curve}
\author{V.Guletski\v{\i}}
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\maketitle
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\section{Introduction}
\label{osina}

Let $k$ be a field, $char(k)\neq 2,3$, and let    
$X$ be a smooth projective geometrically integral curve over $k$. 
Assume that $X$ has at least one point rational over $k$. 
Let $k(X)$ be the field of $k$-rational functions on $X$. 
The Brauer group $\Br (X)$ of $X$ is naturally isomorphic to 
the unramified Brauer group $\Br _{nr}(k(X)/k)$ of $k(X)/k$ 
(see \cite{Lichtenbaum}, \cite{colliot}). This allows to 
identify $\Br (X)$ with $\Br _{nr}(k(X)/k)$. Denote by $J$ the 
Jacobian variety of the curve $X$. In \cite{GMY} the description 
of the $2$-torsion part in $\Br (X)$ in terms of quaternion algebras 
was made under the condition of $k$-rationality of the $2$-torsion 
subgroup of $J$. In \cite{GY} this result was generated on algebras of 
an arbitrary exponent $n$: if all points of order $n$ on $J$ are rational 
over $k$, then any element of order $n$ in $\Br (X)$ can be presented 
by a tensor product of cyclic algebras depending on the $n$-torsion 
subgroup of $J$. 

Let $X=E$ be an elliptic curve over $k$. In this paper we consider the 
problem of a description of $2$-torsion part in $\Br (E)$ in the case when 
$E$ has only one $k$-rational point of order $2$ on the Jacobian $E$.
Our arguments based on an explicit computation of $1$-cocycles presenting
elements of order $2$ in the group of homogeneous spaces over $E$ and 
a construction of a section for some divisor mapping induced by the 
$2$-torsion subgroup of $E$.

It should be mentioned that in \cite{YM}, \cite{VY} and \cite{V} 
a complete description of the $2$-torsion in $\Br (E)$ was made in the case
of a local field $k$ and all proofs there heavily use the completeness 
property of the ground field $k$. In contrast to this we present a different 
approach which is mainly based on cohomology methods and allows to consider 
an arbitrary ground field $k$, $char(k)\neq 2,3$. On the other hand we
do not answer the questions about triviality and indices of algebras of
exponent $2$ in $\Br (E)$.

Since the characteristic of the ground field $k$ is not $2$ or $3$, 
the elliptic curve $E$ can be defined as the
projective closure of a plane curve defined by an equation
   $$
   y^2=F(x)\; ,
   $$
where $F(x)$ is a cubic polynomial over $k$ with the leading 
coefficient equal to $1$ without multiple roots
in a separable closure $\bar k$ of $k$. Let $\bar E=E(\bar k)$ be
the group of geometric points on $E$ and let $\dv \bar E=\{ O,P,T,S\} $
be the $2$-torsion subgroup of $\bar E$. Here $O$ is the
infinite point on $E$ playing the rule of zero on the curve $E$.
Let $x_P$, $x_T$, $x_S$ be the $x$-coordinates of the points $P$, $T$,
$S$ respectively. Then we have the decomposition 
   $$
   F(x)=(x-x_P)(x-x_T)(x-x_S)
   $$
over $\bar k$. We say that $E$ is {\it split} if all $2$-torsion 
points are rational over $k$. This is equivalent to the condition 
that all roots $x_P$, $x_T$, $x_S$ of the polynomial $F(x)$ lie in $k$. 
We say that $E$ is {\it semisplit} if there exists only one $k$-point
in $\dv \bar E$ denoted below by $P$. In other words, only one root $x_P$ of
$F(x)$ is $k$-rational. If all non-trivial points in $\dv \bar E$
are not $k$-rational, then we say that $E$ is {\it fully
non-split}.

In the split case it is not hard to compute quaternion algebras
generating $\dv \Br (E)$. Recall the result of \cite{GY}
for the group $\dv \Br (E)$. Let $E$ be split. Fix any two non-trivial 
$2$-torsion points on $\bar E$, for example $P$ and $T$. Then any 
unramified central simple algebra of exponent $2$ over $k(E)$ similar 
to a tensor product
   $$
   A\otimes (b,x-x_P)\otimes (c,x-x_T)\; .
   $$
Here $A$ is a {\it numerical} algebra over $k(E)$, i.e. an algebra
obtained by the scalar extension of some algebra with the center $k$.
The algebras $(b,x-x_P)$, $(c,x-x_T)$ are quaternion algebras over $k(E)$
with $b,c\in k^*$. Note that $(b,x-x_P)$ or $(c,x-x_T)$ 
can be trivial in $\Br (E)$. For example, if $b$ is a square in $k$ 
then $(b,x-x_P)$ is zero in $\Br (E)$.

If $E$ is fully nonsplit, then the problem of finding of
all unramified central simple algebras of exponent $2$ over $k(E)$
can be reduced to either a split or semisplit case. Indeed, fully
nonsplitness means that $F$ is an irreducible polynomial over $k$.
Then the extension $L=k(x_T)$ has degree $3$ over $k$. Let
$E_L=E\times \Spec (L)$. The composition
   $$
   \Br(E)\stackrel{\res}{\lra }
   \Br(E_L)\stackrel{\cor}{\lra }\Br (E)
   $$
coincides with the multiplication-by-$3$ homomorphism on the group $\Br (E)$. 
So keeping in mind that $2$ and $3$ are relatively prime we get that the 
corestriction map
   $$
   \cor :\dv \Br (E_L)\lra \dv \Br (E)
   $$
is surjective. Hence one can find algebras in $\dv \Br (E)$ applying 
$\cor $ to algebras from $\dv \Br (E_L)$.

Thus we see that the main situation is when $\dv \Br (E)$ has only
one non-trivial point $P$ defined over $k$. In this case
   \begin{equation}
   \label{fom}
   F(x)=(x-x_P)f(x)
   \end{equation}
where $f(x)$ is an irreducible quadratic polynomial over $k$. 

Let $G=\Gal (\bar k/k)$ be the absolute Galois group of the ground
field $k$, $L=k(x_T)=k(x_S)$ and $H=\Gal(\bar k/L)$. Let $U$ be
a subgroup of index $2$ in $H$ and $Z=\bar k^U$. If $U$ is normal 
in $G$ and $G/U$ is a cyclic group of order $4$, then there exists 
a cyclic algebra $(Z\cdot k(E)/k(E),f(x))$ over $k(E)$, where the 
quadratic polynomial $f(x)$ is considered as a rational function on $E$.
If $U$ is not normal in $G$ (this is equivalent to the existing of 
an element $g_0\in G\backslash H$ such that $U\neq g_0^{-1}Ug_0$) 
then one can construct a special central simple algebra $B_U$ over $k(E)$ 
(see Section \ref{meteor} below), which is a cross product defined by 
a $2$-cocycle in an explicit form. We prove the following theorem, 
which is the main result of this paper.

   \begin{theorem}
   \label{main}
   Let $k$ be a field, $char(k)\neq 2,3$, 
   and let $E$ be a semisplit elliptic curve over $k$. Then any element 
   of order $2$ in the quotient group $\Br (E)/\Br (k)$ is similar 
   to a simple algebra of one of the following three types:

   (i)
       $$
       (a,x-x_P)\; ,
       $$
   where $(a,x-x_P)$ is a quaternion algebra over $k(E)$ with $a\in k^*$;

   (ii)
      $$
      (Z\cdot k(E)/k(E),f(x))\; ,
      $$
   where $(Z\cdot k(E)/k(E),f(x))$ is a cyclic algebra over $k(E)$ generated
   by a cyclic extension $Z/k$ of degree $4$ over $k$ such that
   $L\subset Z$ and the polynomial $f$ is defined by 
   (\ref{fom}) and is viewed as a $k$-rational function on $E$;

   (iii)
   an algebra $B_U$, which is defined in
   Section \ref{meteor} by a $2$-cocycle in an explicit form.
   \end{theorem}

Note that the direct computation (see Proposition \ref{atmosfera} below)
shows that the algebra $B_U$ from (iii) has index at most $4$. This agrees 
with the results of Van den Bergh (see \cite{van den Bergh}, Prop. 1, p. 196) 
and implies that $B_U$ is similar to either a quaternion or biquaternion 
algebra over $k(E)$. After the paper was finished V. Chernousov inform me 
that $B_U$ can be decomposed in a tensor product of two explicit quaternion 
algebras determined by $f(x)$ and the subgroup $U$. The correspoding theorem 
can be formulated in the following way:

   \begin{theorem}
   \label{che}
   Let $k$ be a field, $char(k)\neq 2,3$, and 
   $E$ be a semisplit elliptic curve over $k$. Without loss of generality 
   the curve $E$ can be defined by the equation $y^2=(x-p)(x^2-a)$, 
   where $p,a\in k$ and $a$ is not a square in $k$ (hence $f(x)=x^2-a$ 
   and $L=k(\sqrt {a})$). Let $B_U$ be the algebra from (iii) 
   of Theorem \ref{main} and $b=u+v\sqrt {a}$ an element in $L$ such that 
   $\bar k^U=L(\sqrt {b})$. Then $B_U$ is similar to the tensor product 
       $$
       (v,u^2-v^2a)\otimes (vx+u,(u^2-v^2a)(x^2-a))\; ,
       $$    
   where $(v,u^2-v^2a)$ is the numerical quaternion algebra defined by 
   the two constants $v,u^2-v^2a\in k\subset k(E)$ and 
   $(vx+u,(u^2-v^2a)(x^2-a))$ is the quaternion algebra defined by the 
   functions $vx+u,(u^2-v^2a)(x^2-a)\in k(E)$.
   \end{theorem}

\vspace{4mm}

\centerline{\it Acknowledgements}

\vspace{4mm}

This work was supported by the Institute of Mathematics of the
Academy of Sciences of Belarus and TMR ERB FMRX CT-97-0107.
The author thanks  Vladimir Chernousov for useful discussions.

\vspace{4mm}

\centerline{\it Notations}

\vspace{4mm}

Symbols $\bZ $, $\bQ $ denotes integers and rational numbers
respectively. If $A$ is an abelian group and $n$ is a natural
number, then $n:A\ra A$ denotes the homomorphism of multiplication by
$n$ and $_nA$, $A/n$ are its kernel and cokernel respectively. 
For any finite set $S$ we denote the number of elements in $S$ by $\# S$.
If $H$ is a group, $U$ its normal subgroup and
$x\in H$, we denote the left coset $xU$ by $\overline x$.
$F^*$ denotes the multiplicative group of a field $F$. Variety is always
smooth projective and geometrically integral scheme over $k$. For
any variety $X$ over $k$ and for any field extension $L/k$ let
$X(L)$ be the set of $L$-points on $X$ and let $\bar X=X(\bar k )$ be the
set of geometrical points on $X$. A curve is a variety of
dimension one. A point of a curve means closed point. 
We denote an arbitrary curve by $X$. An elliptic curve will 
be always denoted by $E$.

\section{Preliminary results}
\label{africa}

In this section we recall some well-known facts used below. All of
them can be found in \cite{Fadd}, \cite{Milne} and
\cite{Scharlau}.

\subsection{\it Short exact sequences}
\label{sosna}

Let $G=\Gal (\bar k/k)$ be the absolute Galois group of the ground
field $k$, $X$ a curve of an arbitrary genus defined over $k$
such that $X(k)\neq \emptyset $ and let $\bar k(\bar X)$ be the field of
$\bar k$-rational functions on $\bar X$. One can associate with the 
curve $X$ the following discrete (left) $G$-modules:
$\Div(\bar X)$ is the divisor group of $\bar X$; 
$\Div^0(\bar X)$ is its subgroup of degree zero divisors on $\bar X$;
$\Pri(\bar X)$ is the subgroup of principles divisors;
$\Pic(\bar X)=\Div(\bar X)/\Pri(\bar X)$ and $\Pic^0(\bar X)=\Div^0(\bar
X)/\Pri(\bar X)$. These $G$-modules are terms of the following short
exact sequences

\begin{equation}
\label{gldiv} 1\ra \bar k^*\lra \bar k(\bar X)^*
\stackrel{\rm{div}}{\lra} \Pri(\bar X)\ra 0\; ,
\end{equation}
\begin{equation}
\label{pic} 0\ra \Pri(\bar X)\lra \Div(\bar X) \lra \Pic(\bar
X)\ra 0\; ,
\end{equation}
\begin{equation}
\label{picnol} 0\ra \Pic^0(\bar X)\lra \Pic(\bar X)
\stackrel{\deg}{\lra} \bZ\ra 0\; .
\end{equation}
Here $\deg $ is the degree homomorphism and $\di $ is the homomorphism
sending $f\in \bar k(\bar X)^*$ into the divisor $\di (f)$ of zeros 
and poles of the rational function $f$. Let
$\bar J$ be the group of $\bar k$-points of the Jacobian variety
$J$ of the curve $X$. Since we suppose that $X$ has at
least one $k$-rational point, $\bar J$ and $\Pic ^0(\bar X)$ are
isomorphic as discrete $G$-modules. Therefore we have
the short exact sequence
\begin{equation}
\label{ji} 0\ra \Pri(\bar X)\lra \Div^0(\bar X)
\stackrel{\rm{sum}}{\lra} \bar J\ra 0\; .
\end{equation}

\subsection{\it Local invariants}
\label{siren'}

Denote the set of closed points on $X$ (i.e. the set of points of codimension 
1 on $X$) by $X^{(1)}$. All such points $x\in X^{(1)}$
are in one-to-one correspondence with discrete valuations rings of
the field $k(X)$ over $k$. Let $k(X)_x$ be the completion of
$k(X)$ with respect to the valuation corresponding to a point $x\in
X^{(1)}$. Denote by $k(x)$ the residue field of $X$ in $x$. Then
$[k(x):k]<\infty $ and one can identify $k(x)$ with a subfield in
$\bar k$. Let $G_x=\Gal (\bar k/k(x))$ for any $x\in X^{(1)}$.
Then $\Br (k(X)_x)$ is isomorphic to the direct sum of the
Brauer group $\Br (k(x))$ and the group $\Hom _{cont}(G_x,\bQ /\bZ
)$ of continuous homomorphisms of the profinite group $G_x$ into
the discrete group $\bQ /\bZ $. Let
    $$
    \inv _x:\Br(k(X))\lra \Hom _{cont}(G_x,\bQ /\bZ )
    $$
be the composition of the scalar extension homomorphism
    $$
    \Br(k(X))\lra \Br(k(X)_x)
    $$ 
and the projection 
    $$
    \Br (k(X)_x)\cong \Br (k(x))\oplus 
    \Hom _{cont}(G_x,\bQ /\bZ )\ra \Hom _{cont}(G_x,\bQ/\bZ )\; .
    $$ 
The homomorphism $\inv _x$ maps a class of algebras $a\in
\Br(k(X))$ into its {\it local invariant} $\mbox{inv}_x(a)$.
The order of $\mbox{inv}_x(a)$ coincides with the ramification index
of algebras from $a$ at the point $x\in X^{(1)} $. Algebras can be
ramified only in a finite number of points. Hence one can construct
the homomorphism
    $$
    \mbox{inv}:\Br(k(X))\lra \bigoplus _{x\in X^{(1)}}
    \Hom _{cont}(G_x,\bQ /\bZ )\; .
    $$
By definition, its kernel is {\it unramified Brauer group} $\Br
_{nr}(k(X))$ of $k(X)/k$. As it was mentioned in the beginning,
the group $\Br _{nr}(k(X))$ can be naturally identified with the
Brauer group $\Br(X)$ of the curve $X$.

Recall the cohomology description of the homomorphism $\inv $.
Below $H^*(G,?)$ is the Galois cohomology functor on the
category of (left) discrete $G$-modules. By Tsen theorem, the group
$\Br(k(X))$ is isomorphic to the cohomology group $H^2(G,\bar
k(\bar X)^*)$. Let $M_x$ be a sumbmodule in $\Div(\bar X)$
generated by points in $\bar X$ mapping into $x\in X^{(1)}$ under the natural
projection $\bar X\ra X$. The discrete $G$-module $M_x$ is identified
with the induced $G$-module $M_G^{G_x}(\bZ)$, where $\bZ $ is viewed as
a trivial $G$-module. Then $\Div(\bar X)\cong \bigoplus _xM_G^{G_x}(\bZ)$, 
where $x$ runs all points in $X^{(1)}$. Thus we get
    $$
    H^2(G,\Div(\bar X))\cong
    \bigoplus _xH^2(G,M_G^{G_x}(\bZ))
    \cong \bigoplus _xH^2(G_x,\bZ )
    \cong \bigoplus _x\Hom _{cont}(G_x,\bQ /\bZ )\; .
    $$
It can be proven that the composition
    $$
    \Br(k(X))\cong H^2(G,\bar k(\bar X)^*)
    \stackrel{{\di }^*}{\lra} H^2(G,\Div(\bar X))\cong
    \bigoplus _x\; \Hom _{cont}(G_x,\bQ /\bZ )
    $$
coincides with the homomorphism $\inv $. Here ${\di }^*$ is the
homomorphism induced by $\di $ on cohomology groups. Therefore,
    $$
    \Br(X)\cong \ker ({\di }^*)\; .
    $$

\subsection{\it Principle homogeneous spaces}
\label{lebeda}

Consider the exact sequence
    $$
    H^2(G,\bar k^*)\ra H^2(G,\bar k(\bar X)^*)
    \stackrel{\alpha}{\ra}
    H^2(G,\Pri(\bar X))\ra H^3(G,\bar k^*)
    \ra H^3(G,\bar k(\bar X)^*)
    $$
induced by (\ref{gldiv}). Since $X(k)\neq \emptyset $, the left
and right homomorphisms are injective for any $n$ (see \cite{Scharlau}).
Hence we have the short exact sequence
    $$
    0\ra H^2(G,\bar k^*)\lra H^2(G,\bar k(\bar X)^*)
    \stackrel{\alpha}{\lra} H^2(G,\Pri(\bar X))\ra 0\; .
    $$
Taking into account the isomorphisms $H^2(G,\bar k^*)\cong \Br(k)$ and
$H^2(G,\bar k(\bar X)^*)\cong \Br(k(X))$, one can obtain the exact sequence
    $$
    0\ra \Br(k)\stackrel{\iota}{\lra}
    \Br(k(X)) \stackrel{\alpha}{\lra}
    H^2(G,\Pri(\bar X)) \ra 0\; ,
    $$
where $\iota $ is induced by the scalar extension: if $A$ is a simple
algebra with the center $k$, then $\iota ([A])= [A\otimes _kk(X)]$.

Now observe that ${\di }^*=\beta \circ \alpha $,
where $H^2(G,\Pri(\bar X))\stackrel{\beta}{\ra} H^2(G,\Div(\bar
X))$ is the homomorphism induced by the embedding $\Pri(\bar
X)\subset \Div(\bar X)$. Since $\alpha $ is surjective, one has
$\mbox{ker}({\di }^*)=\alpha ^{-1}(\mbox{ker}(\beta ))$. Consider
the exact sequence
    $$
    H^1(G,\Pic(\bar X))\stackrel{\Delta} {\lra}
    H^2(G,\Pri(\bar X))\stackrel{\beta}{\lra} H^2(G,\Div(\bar X))
    $$
induced by (\ref{pic}). It follows that $\mbox{ker}({\di
}^*)=\alpha ^{-1}(\mbox{im}(\Delta))$. Furthermore, consider the exact
sequence
    $$
    \Pic(\bar X)^G\stackrel{\deg}{\lra} \bZ \ra
    H^1(G,\Pic^0(\bar X))\ra H^1(G,\Pic(\bar X))\ra 0
    $$
induced by (\ref{picnol}). Here $\Pic(\bar X)^G$ is the group of
divisor classes invariant under the action of $G$. The last sequence
is ended by zero because $H^1(G,\bZ )=0$. Since $X(k)\neq \emptyset $,
the homomorphism $\deg$ is surjective, hence the epimorphism
$H^1(G,\Pic^0(\bar X))\ra H^1(G,\Pic(\bar X))$ is injective. Thus
we have that
    $$
    H^1(G,\Pic(\bar X))\cong H^1(G,\Pic^0(\bar X))\cong H^1(G,\bar J)\; .
    $$
Then
    $$
    \mbox{ker}({\di }^*)=\alpha ^{-1}(\mbox{im}(\delta ))\; ,
    $$
where $\delta :H^1(G,\bar J)\ra H^2(G,\Pri(\bar X))$ is the connecting
homomorphism induced by (\ref{ji}). Note that the condition $H^1(G_x,\bZ )=0$
implies that $H^1(G,\Div(\bar X))=0$. Then the exact sequence
induced by (\ref{pic}) yields that the connecting homomorphism $\Delta$
is injective. Hence $\delta $ is injective too. Let
    $$
    \delta ^{-1}:\im(\delta )\ra H^1(G,\bar J)
    $$
be the inverse to the isomorphism $\delta :H^1(G,\bar J)\ra \im(\delta
)$. Define $\pi $ as the composition
    \begin{equation}
    \label{tiski}
    \pi :\Br(X)\cong \mbox{ker}({\di }^*) \stackrel{\alpha }{\lra}
    \im(\delta ) \stackrel{\delta ^{-1}}{\lra}H^1(G,\bar J)\; .
    \end{equation}
The definition of $\pi $ can be illustrated by the following commutative
diagram:
    $$
    \xymatrix{
    & & & & H^2(G,\Div ^0(\bar X))
    \ar@{^{(}->}[d] \\
    H^1(G,\bar J)
    \ar@{^{(}->}[rr]^-{\delta } & &
    H^2(G,\Pri (\bar X))
    \ar[rru] \ar[rr]_-{\beta } & &
    H^2(G,\Div (\bar X)) \\
    \ker (\inv )
    \ar[u] \ar@{^{(}->}[rr] & &
    H^2(G,\bar k(\bar X)^*)
    \ar@{>>}[u]_-{\alpha }
    \ar[rru]_-{\inv } & & \\
    \Br (X)
    \ar[u]^-{\cong }
    \ar@{^{(}->}[rr] & &
    \Br (k(X)) \ar[u]^-{\cong }
    }
    $$

Now we have the exact sequence
    \begin{equation}
    \label{maikop}
    0\ra \Br(k)\stackrel{\iota}{\lra} \Br(X)\stackrel{\pi}{\lra}
    H^1(G,\bar J)\ra 0\; .
    \end{equation}
It splits. Indeed, if $x\in X(k)$, then $k(x)=k$ and $\Br
(k(X)_x)\cong \Br (k)\oplus Hom_{cont}(G,\bQ /\bZ )$. One can
check that the composition
    $$
    \Br(k)\ra \Br(k(X))\ra \Br(k(X)_x)\ra
    \Br(k)\oplus Hom_{cont}(G,\bQ /\bZ ) \ra \Br(k)
    $$
is an identical map. Hence $\iota $ has a left inverse.

\subsection{\it Torsion}
\label{krapiva}

Let $n$ be a natural number. If $A\stackrel{\iota}{\lra} B$ is a
homomorphism of abelian groups, then denote by
$_nA\stackrel{_n\iota}{\lra} \, _nB$ and $A/n\stackrel{\iota
/n}{\lra} B/n$ the homomorphisms induced by $\iota $. The sequence
(\ref{maikop}) induces the exact sequence
    $$
    0\ra\, _n\Br(k)\stackrel{_n\iota}
    {\lra}\, _n\Br(X) \stackrel{_n\pi}{\lra}\, _nH^1(G,\bar J)\ra
    \Br(k)/n\stackrel{\iota /n}{\lra} \Br(X)/n
    \stackrel{\pi /n}{\lra}H^1(G,\bar J)/n \ra 0\; .
    $$
If $\hat{\iota }$ is a left inverse to $\iota $, then $\hat{\iota
}/n$ is a left inverse to $\iota /n$. Therefore $\iota /n$ is
injective. Then the last exact sequence yields the short exact
sequence
   $$
   0\ra\, _n\Br(k) \stackrel{_n\iota}{\lra}\, _n\Br(X)
   \stackrel{_n\pi}{\lra}\, _nH^1(G,\bar J)\ra 0\; .
   $$
We shall use the following simple
   \begin{lemma}
   \label{sech}
   Let $0\rightarrow A\stackrel{\iota}{\longrightarrow}
   B \stackrel{\pi}{\longrightarrow} C\rightarrow 0$ be an
   exact sequence. Suppose that the homomorphism $\iota $
   has a left inverse $\hat \iota :B\rightarrow C$. For any
   $c\in C$ choose $b_c\in \pi ^{-1}(c)$. Then any element
   $b\in B$ has the unique decomposition $b=b_c+\iota (a)$, where
   $c=\pi (b)\in C$ and $a=\iota ^{-1}(b-b_c)$. The map $\hat \pi :
   C\rightarrow B$, $c\mapsto b_c-\iota \hat \iota (b_c)$ is a
   right inverse homomorphism of $\pi $.
   \end{lemma}

\subsection{\it The plane of computations}
\label{plane}

Now let $X=E$ be an elliptic curve over $k$. Then $\bar J=\bar E$ and
there exists the split exact sequence
    $$
    0\ra\, _2\Br(k)
    \stackrel{_2\iota}{\lra}\, _2\Br(E) \stackrel{_2\pi}{\lra}\,
    _2H^1(G,\bar E)\ra 0\; .
    $$
For every cohomology class $c\in \dv H^1(G,\bar E)$ we want to
choose a Brauer class $b_c\in \dv \Br (E)$ such that $\dv \pi (b_c)=c$. 
We shall describe classes $c\in \dv H^1(G,\bar E)$ in terms of explicit
cocycles (Section \ref{kronshtadt}). After that we construct
central simple algebras representing the Brauer classes $b_c$ in $\Br (E)$ 
(Sections \ref{kamchatka} and \ref{dihedral}). Since by Lemma \ref{sech}
any element $b\in \dv \Br (E)$ has a unique
decomposition $b=\dv \iota (a)+b_c$, where $a\in \dv \Br (k)$ and
$c\in \dv H^1(G,\bar E)$, we get that
a full description of 2-torsion part of $\Br (E)/\Br (k)$ is given 
by central simple algebras representing classes $b_c$.

The Kummer sequence
    \begin{equation}
    \label{kumm}
    0\ra \dv \bar E\lra \bar E\stackrel{2}{\lra} \bar E\ra 0
    \end{equation}
yields the exact sequence
    \begin{equation}
    \label{batyi}
    0\ra E(k)/2\lra H^1(G,\dv \bar E)\lra \dv H^1(G,\bar E)\ra 0\; .
    \end{equation}
It follows that any element in $\dv H^1(G,\bar E)$ can be
presented by a cocycle $\phi :G\ra \dv \bar E$ of the group $G$
with coefficients in $\dv \bar E$. Moreover, the cohomology class
$\cls (\phi )\in \dv H^1(G,\bar E)$ is trivial iff the cohomology
class $\cls (\phi )\in H^1(G,\dv \bar E)$ lies in the image of the
embedding $E(k)/2\hra H^1(G,\dv \bar E)$. This suggests that
firstly we need to describe all $1$-cocycles of the group $G$ with
coefficients in $\dv \bar E$. It will provide the description of
$\dv H^1(G,\bar E)$ in terms of explicit $1$-cocycles of the group
$G$ with coefficients in $\dv \bar E$.

\section{Calculus of cocycles}
\label{kronshtadt}

Let $E$ be a semisplit elliptic curve and let $\{ O,P\} $ be the 
group $k$-rational $2$-torsion points on $E$. Then $F(x)=(x-x_P)f(x)$, 
where $f(x)$ is an irreducible quadratic polynomial over $k$. 
If $\dv E=\{ O,P,T,S\} $ is the group of all $2$-torsion points on 
$\bar E$ then $f(x)=(x-x_T)(x-x_S)$ over $\bar k$. Let $L/k$ be the minimal
subextension in $\bar k/k$ such that all $2$-torsion points of $E$
are rational over $L$, i.e. $L=k(x_T)=k(x_S)$. The group $H=\Gal (\bar k/L)$
has index $2$ in $G$.

\subsection{\it Starting lemmas}
\label{uhnov}

Consider the exact sequence
   $$
   0\ra H^1(G/H,\dv \bar E)\stackrel{\inf }{\lra }H^1(G,\dv \bar E)
   \stackrel{\inf }{\lra }H^1(H,\dv \bar E)^{G/H}\lra H^2(G/H,\dv \bar E)
   $$
induced by the Hochschild-Serre spectral sequence $H^p(G/H,H^q(H,\dv
\bar E ))\Rightarrow H^n(G,\dv \bar E)$ (see, for example,
\cite{Serre}). It is easy to show that $H^1(G/H,\dv \bar
E)=0$ and $H^2(G/H,\dv \bar E)=0$. So we have the isomorphism
   \begin{equation}
   \label{vyaz'ma}
   \res :H^1(G,\dv \bar E)\stackrel{\cong }{\lra }H^1(H,\dv \bar
   E)^{G/H}\; .
   \end{equation}
Recall the action of $G/H$ on $H^1(H,\dv \bar E)$. Let $\psi \in
H^1(H,\dv \bar E)=Hom(H,\dv \bar E)$ and $g_0\in G\backslash H$.
Then $(g_0\psi )(g)=g_0(\psi (g_0^{-1}gg_0))$ for any $g\in G$.
Note that this action does not depend on the choice of $g_0$ and we
can consider the action of $g_0$ as an action of the unique
non trivial element in $G/H$.

Now fix an isomorphism
   $$
   \dv \bar E\cong \{ O,T\}\oplus \{ O,S\} \; .
   $$
Then
   $$
   Hom(H,\dv \bar E)\cong
   Hom(H,\{ O,T\} )\oplus Hom(H,\{ O,S\})\; .
   $$
Let $(\psi _1,\psi _2)$ be an element in $Hom(H,\{ O,T\} )\oplus
Hom(H,\{ O,S\})$ and $U_i=\ker (\psi _i)$, $i=1,2$. Then the pair
$(\psi _1,\psi _2)$ lies in $(Hom(H,\{ O,T\} )\oplus Hom(H,\{
O,S\}))^{G/H}$ if and only if $U_2=g_0^{-1}U_1g_0$ for some
$g_0\in G\backslash H$ (this condition does not depend on the choice
of an element $g_0$ in $G\backslash H$). Therefore one can
identify the set of nontrivial homomorphisms in $(Hom(H,\{ O,T\}
)\oplus Hom(H,\{ O,S\}))^{G/H}$ with the set of subgroups $U$ of
index two in $H$. A such $U$ corresponds to the homomorphism $\psi
=\psi _T+\psi _S$, where $\psi _T$ is the surjective homomorphism
$H\ra \{ O,T\} $ with the kernel $U$, and $\psi _S$ is the surjective
homomorphism $H\ra \{ O,S\} $ with the kernel $g_0^{-1}Ug_0$.

   \begin{lemma}
   \label{opushka}
   If $\phi :G\ra \dv \bar E$ is a cocycle, then
   $V=\ker(\phi \palka _H)$ is a normal subgroup in $G$.
   \end{lemma}
\begin{pf}
Isomorphism (\ref{vyaz'ma}) shows that $\phi \palka _H$ is invariant
under the action of $G/H$. Then either $V=H$ or $V=U\cap
g_0^{-1}Ug_0$ for some subgroup $U$ of index $2$ in $H$. In the
second case $g_0^{-1}Vg_0=g_0^{-1}(U\cap
g_0^{-1}Ug_0)g_0=g_0^{-1}Ug_0\cap g_0^{-2}Ug_0^2=g_0^{-1}Ug_0\cap
U=V$. For any element $g\in G$ one can write $g=hg_0$ for some $h\in H$.
Then $g^{-1}Vg= g_0^{-1}h^{-1}Vhg_0=g_0^{-1}Vg_0\subset V$.
\end{pf}
   \begin{lemma}
   \label{prud}
   If $\phi :G\ra \dv \bar E$ is a cocycle and $\psi =\phi \palka H$,
   then $\psi (g^2)\in \{ O,P\} $ for any $g\in G\backslash H$.
   \end{lemma}
\begin{pf}
Since $\psi \in H^1(H,\dv \bar E)^{G/H}$, it follows that $g_0\psi (h)=\psi
(g_0^{-1}hg_0)$ for any $h\in H$. Substituting $h=g^2$ we get
   $$
   \begin{array}{rcl}
   g_0\psi (g^2)
   &=&
   \psi (g_0^{-1}g^2g_0)=\psi (g_0^{-1}g)+\psi (gg_0)\\
   &=&
   \psi (gg_0)+\psi (g_0^{-1}g)\\
   &=&
   \psi (gg_0g_0^{-1}g)\\
   &=&
   \psi (g^2) \\
   &\Lra &
   \psi (g^2)\in \{ O,P\} \; .
   \end{array}
   $$
\end{pf}

\subsection{\it Formula for cocycles}
\label{aurora}

Now let $g_0$ be some element in $G\backslash H$. For any $Q\in
\dv \bar E=\{ O,P,T,S\} $ and for any homomorphism $\psi \in
Hom(H,\dv \bar E)$ denote by $\phi _{Q,\psi }$ the map defined by
the rule
    \begin{equation}
    \label{trostnik}
    \phi _{Q,\psi }(g)=\left\{
    \begin{array}{ll}
    \psi (g)\, , & g\in H \\
    g_0\psi (g_0^{-1}g)+Q\, , &
    g\in G\backslash H\; .
    \end{array}
    \right.
    \end{equation}

   \begin{lemma}
   \label{okolica}
   Any cocycle $\phi :G\ra \dv \bar E$ is of the form $\phi _{Q,\psi
   }$ for some $Q\in \dv \bar E$ and $\psi \in Hom(H,\dv \bar E)$.
   Conversely, let $Q$ and $\psi $ be as above. Then $\pp \phi _{Q,\psi }=0$
   if and only if the following two conditions hold:

   (i) $g_0\psi (h)=\psi(g_0^{-1}hg_0)$ for any $h\in H$;

   (ii) $\psi (g_0^2)=P$ for $Q\in \{ T,S\} $ and $\psi (g_0^2)=O$ for
        $Q\in \{ O,P\} $.
   \end{lemma}
\begin{pf}
If $\phi $ is a $1$-cocycle with coefficients in $\dv \bar E$ then
for any $g\in G$ we have
    $$
    \pp \phi (g_0,g_0^{-1}g)=g_0\phi (g_0^{-1}g)+\phi (g)+\phi (g_0)=0
    $$
for some $g_0\in G\backslash H$. Therefore $\phi =\phi _{Q,\psi
}$, where $Q=\phi (g_0)$ and $\psi =\phi \palka _H$.

If $h_1,h_2\in H$, then it is clear that $\pp \phi _{Q,\psi
}(h_1,h_2)=0$. Let $g\in G\backslash H$ and $h\in H$. Then
  $$
  \begin{array}{rcl}
  \pp \phi _{Q,\psi }(g,h)
  &=&
  g\psi (h)+g_0\psi (g_0^{-1}gh)+Q+ g_0\psi (g_0^{-1}g)+Q \\
  &=&
  g\psi (h)+g_0\psi (g_0^{-1}g)+g_0\psi (h)+ g_0\psi (g_0^{-1}g) \\
  &=&
  g\psi (h)+g_0\psi (h) \\
  &=&
  \left\{
  \begin{array}{ll}
  2\psi (h)\, , & \mbox{�᫨ $\psi (h)\in \{ O,P\} $} \\
  2g_0\psi (h)\, , & \mbox{�᫨ $\psi (h)\in \{ T,S\}$}
  \end{array}
  \right.
  \\
  &=&
  0
  \end{array}
  $$
Note that we replace signs "$-$" by "$+$" because $\phi _{Q,\psi
}$ is a map into a $2$-torsion group.

If $h\in H,g\in G\backslash H$, then for any $Q$ and $\psi $ 
we have
  $$
  \begin{array}{rcl}
  \pp \phi _{Q,\psi }(h,g)
  &=&
  g_0\psi (g_0^{-1}g)+Q+g_0\psi (g_0^{-1}hg)+Q+\psi (h)\\
  &=&
  g_0\psi (g_0^{-1}g)+g_0\psi (g_0^{-1}hg_0)+
  g_0\psi (g_0^{-1}g)+\psi (h) \\
  &=&
  g_0\psi (g_0^{-1}hg_0)+\psi (h)\; .
  \end{array}
  $$
Therefore, if $\pp \phi _{Q,\psi }=0$, then $g_0\psi
(g_0^{-1}hg_0)+\psi (h)=0$, that is
   \begin{equation}
   \label{nabludatel'}
   g_0\psi (h)=\psi (g_0^{-1}hg_0)\; .
   \end{equation}

If $g_1\in G\backslash H,g_2\in G\backslash H$, then for any $Q$
and $\psi $ we have:
   $$
   \begin{array}{rcl}
   \pp \phi _{Q,\psi }(g_1,g_2)
   &=&
   g_1(g_0\psi (g_0^{-1}g_2)+Q)+\psi (g_1g_2)+g_0\psi (g_0^{-1}g_1)+Q
   \\
   &=&
   \psi (g_0^{-1}g_2)+\psi (g_1g_2)+g_0\psi (g_0^{-1}g_1)+g_1Q+Q
   \\
   &\stackrel{(\ref{nabludatel'})}{=}&
   \psi (g_0^{-1}g_2)+\psi (g_1g_2)+
   \psi (g_0^{-1}g_0^{-1}g_1g_0)+g_1Q+Q \\
   &=&
   \psi (g_0^{-1}g_2)+\psi (g_1g_2)+\psi (g_0^2)+\psi
   (g_1g_0)+g_1Q+Q\; .
   \end{array}
   $$
Observe that
   $$
   \begin{array}{rcl}
   \psi (g_0^{-1}g_2)+\psi (g_1g_2)+\psi (g_1g_0)
   &=&
   \psi (g_1g_0)+\psi (g_0^{-1}g_2)+\psi (g_1g_2)\\
   &=&
   \psi (g_1g_0g_0^{-1}g_2)+\psi (g_1g_2)\\
   &=&
   \psi (g_1g_2)+\psi (g_1g_2)\\
   &=&
   2\psi (g_1g_2)\\
   &=&
   0
   \end{array}
   $$
Hence one can continue:
   $$
   \begin{array}{rcl}
   \pp \phi _{Q,\psi }(g_1,g_2)
   &=&
   \psi (g_0^{-1}g_2)+\psi (g_1g_2)+\psi (g_0^2)+\psi
   (g_1g_0)+g_1Q+Q\\
   &=&
   \psi (g_0^2)+g_1Q+Q\\
   &=&
   \psi (g_0^2)+g_0Q+Q\; .
   \end{array}
   $$
Therefore, if $\pp \phi _{Q,\psi }=0$, then
   $$
   \psi (g_0^2)=g_0Q+Q\; .
   $$
\end{pf}

\begin{remark}
Clearly, the condition $(i)$ is equivalent to the condition
$\psi \in (Hom(H,\dv \bar E))^{G/H}$ and can be deduced directly
from the isomorphism \ref{vyaz'ma}.
\end{remark}

\subsection{\it Classification of cocycles}
\label{kwazar}

Now one can classify all $1$-cocycles $\phi =\phi _{Q,\psi }$ of the
group $G$ with coefficients in $\dv \bar E$ according to the
number of elements in $\im (\psi )$.

Denote by $\xi $ the surjective homomorphism $G\ra \{ O,P\} $ with
kernel $H$. Then $\xi =\pp T=\pp S$ and $\xi $ is a unique
non-trivial $1$-coboundary of $G$ with coefficients in $\dv \bar E$.
If $\#\im (\psi )=1$, i.e. $\psi =0$, then the formula \ref{trostnik}
and Lemma \ref{okolica} show that either $\phi =0$ or $\phi =\xi
$. In both cases $\cls(\phi )=0$ in $H^1(G,\dv \bar E)$.

Let $\phi _{Q,\psi }$ be a cocycle. Let $\psi _T$ and $\psi _S$
are the compositions of $\psi $ with the projections of the group $\dv
\bar E$ on $\{ O,T\} $ and $\{ O,S\} $ respectively. 
Let $U=\ker (\psi _T)$. Then $\ker (\psi _S)=g_0^{-1}Ug_0$. The condition
$\# \im (\psi )=2$ is equivalent to the condition $g_0^{-1}Ug_0=U$, 
$g_0\in G\backslash H$. In this case $\psi $ is
a surjective homomorphism $H\ra \{ O,P\} $ with kernel $U$. The
group $U$ is normal in $G$ by Lemma \ref{opushka}. Consider the
quotient group $G/U$. There are two possible cases for $G/U$:
either $G/U\cong \bZ /2\oplus \bZ /2$ or $G/U\cong \bZ /4$. In
the first case one can choose $g_0\in G\backslash H$ such that
$g_0^2\in U$. By Lemma \ref{okolica} the point $Q$ lie in $\{ O,P\} $. But
$\phi _{P,\psi }+\phi _{O,\psi }=\xi =\pp T$. Consequently, we can
consider only cocycles $\phi _{O,\psi }$. If $G/U\cong \bf Z/4$
then $g_0^2\in H\backslash U$. By Lemma \ref{okolica} the point 
$Q$ lies in $\{ T,S\} $. Adding $\xi $ one can consider only cocycles
$\phi _{T,\psi }$.
   \begin{definition}
   \label{doloto}
   Let $\phi =\phi _{Q,\psi }$ be a cocycle with $\# \im (\psi )=2$
   and $U=\ker (\psi )$. If $G/U\cong \bZ /2\oplus \bZ /2$ and
   $Q=O$, then we say that $\phi $ is a cocycle of
   the type $I$. If $G/U\cong \bZ /4$ and $Q=T$,
   then we say that $\phi $ is a cocycle of the type $II$.
   \end{definition}

Now let us consider the case of a cocycle $\phi =\phi _{Q,\psi }$
with $\# \im (\psi )=4$. The corresponding homomorphism can be
uniquely determined by a subgroup $U$ of index 2 in $H$ such that
$U\neq g_0^{-1}Ug_0$ for some $g_0\in G\backslash H$ (note that
$g_0^{-1}Ug_0={g_0'}^{-1}Ug_0'$ for any two elements $g_0,g_0'\in
G\backslash H$). Let $U_O=U\cap g_0^{-1}Ug_0$. 
The condition $U\neq g_0^{-1}Ug_0$ implies that the quotient
group $G/U_O$ is noncommutative of order $8$. Moreover, $G/U_O$ has
$3$ elements of order $2$ lying in the subgroup $H/U_O$. Therefore 
$G/U_O$ is isomorphic to the dihedral group $D_4$ (see, for example, 
\cite{Hall}, 4.4). But $D_4$ has five elements of order $2$. Consequently,
there exists an element $g_0$ in $G\backslash H$ such that $g_0^2$ 
lies in $U_O$. By Lemma \ref{okolica} the point $Q$ lies $\{ O,P\} $. 
Adding $\xi $ if it is necessary one can consider only the case $Q=O$.
   \begin{definition}
   \label{instrument}
   Cocycles $\phi =\phi _{T,\psi }$ with $\# \im (\psi )=4$ will be
   called cocycles of type $III$.
   \end{definition}

In fact we proved the
   \begin{proposition}
   \label{venera}
   Any element in $\dv H^1(G,\bar E)$ can be represented by
   cocycle of one of three types $I$, $II$ and $III$.
   \end{proposition}

\subsection{\it Direct calculations of cocycles}
\label{pluton}

    \begin{proposition}
    \label{vologda}
    Let $\phi =\phi _{Q,\psi }$ be a cocycle of the group $G$
    with coefficients in $\dv \bar E$ and $U=\ker (\psi )$.
    If $\phi $ is a cocycle of type $I$, then
    $\phi $ is the surjective homomorphism $G\ra \{ O,P\} $ with kernel
    $U\cup g_0U$. If $\phi =\phi _{T,\psi }$ has type $II$, then
    $$
    \phi _{T,\psi }(g)=\left\{
    \begin{array}{llll}
    O\; ,\; \, \hbox{if}\; \, g\in U \\
    T\; ,\; \, \hbox{if}\; \, g\in g_0U \\
    P\; ,\; \, \hbox{if}\; \, g\in H\backslash U \\
    S\; ,\; \, \hbox{if}\; \, g\in G\backslash (H\cup g_0U)
    \end{array}
    \right.
    $$
    \end{proposition}
\begin{pf}
Let $\phi _{O,\psi }$ be a cocycle of type $I$. Then $G/U\cong \bZ
/2\oplus \bZ /2$ and $\im (\psi )=\{ O,P\} $. If $g\in G\backslash
(H\cup g_0U)$, then $g_0^{-1}g\in H\backslash U$. Using formula
(\ref{trostnik}) one can compute $\phi _{O,\psi }(g)=\psi
(g_0^{-1}g)=P$. Moreover, for any $u\in U$ we have $\phi _{O,\psi
}(g_0u)=\psi (g_0^{-1}g_0u)=\psi (u)=O$.

Let $\phi _{T,\psi }$ be a cocycle of type $II$. Fix the
isomorphism $G/U\cong \bZ /4$ such that $\overline {g_0}$ maps
into the $1+\bZ $. Sometimes it is convenient to identify the group $G/U$
and $\bZ /4$ by means of this isomorphism. Any $g\in G$ can be 
written in the form
   $$
   g=g_0^iu\; \, \hbox{where}\; \, i\in \{ 0,1,2,3\}
   \; \, \hbox{and}\; \, u\in U\; ,
   $$
where $i$ is the {\it order} of an element $g$ under the element
$g_0$. If the order $i$ of $g$ is even, then $g_0^i\in
H$. Using formula \ref{trostnik} one can obtain that
   $$
   \phi _{T,\psi }(g)=\psi (g_0^iu)=\psi (g_0^i)=\left\{
   \begin{array}{ll}
   O\; ,\; \, \hbox{for}\; \, i=0 \\
   P\; ,\; \, \hbox{for}\; \, i=2
   \end{array}
   \right.
   $$
If $i$ is odd, then $g_0^i\in G\backslash H$ and hence
$g=g_0^iu\in G\backslash H$. Using the same lemma we have
   $$
   \phi _{T,\psi }(g)=\psi (g_0^{-1}g)+T=\psi (g_0^{-1}g_0^iu)+T
   $$
   $$
   =\psi (g_0^{i-1}u)+T=\psi (g_0^{i-1})+T
   $$
   $$
   =\left\{
   \begin{array}{ll}
   T\; ,\; \, \hbox{for}\; \, i=1 \\
   S\; ,\; \, \hbox{for}\; \, i=3
   \end{array}
   \right.
   $$
\end{pf}

Now let us consider cocycles $\phi _{O,\psi }$ of type $III$.
It follows from the definition that every such cocyle is completely
determined by a subgroup $U$ of index $2$ in $H$ such that $U\neq
g_0^{-1}Ug_0$. Here $g_0\in G\backslash H$ and $g_0^2\in U_O=
U\cap g_0^{-1}Ug_0$. If $\psi _T$ and $\psi _S$ are the compositions
of $\psi $ with the projections of $\dv \bar E$ on $\{ O,T\} $ and
$\{ O,S\} $ respectively, then $\psi =\psi _T+\psi _S$ and $U=\ker
(\psi _T)$, $g_0^{-1}Ug_0=\ker (\psi _T)$ (see Section
\ref{kwazar}). Let
   $$
   U_S=U\backslash (g_0^{-1}Ug_0)
   $$
   $$
   U_T=(g_0^{-1}Ug_0)\backslash U
   $$
   $$
   U_P=H\backslash (U\cup g_0^{-1}Ug_0)
   $$
be the cosets in $H/U_O$. Then $G/U_O=\{ U_O,U_S,U_T,U_P,
g_0U_O,g_0U_S,g_0U_T,g_0U_P,\} $. Schematically one can describe
the quotient group $G/U_O$ by the diagram

\vspace{4mm}

\begin{center}

   \begin{tabular}{|c|c|}
   \hline
   & \\
   $U_S$ & $U_P$ \\
   & \\
   \hline
   & \\
   $U_O$ & $U_T$ \\
   & \\
   \hline
   \hline
   & \\
   $g_0U_S$ & $g_0U_P$ \\
   & \\
   \hline
   & \\
   $g_0U_O$ & $g_0U_T$ \\
   & \\
   \hline
   \end{tabular}

\end{center}

\vspace{4mm}

\noindent Here the top four squares correspond to the group $H$
and all eight squares correspond to the whole group $G$.

    \begin{proposition}
    \label{irkutsk}
    Let $\phi =\phi _{O,\psi }$ be a cocycle
    of type $III$. Let
    $$
    V_O=U_O\cup g_0U_O\; ,
    $$
    $$
    V_S=U_S\cup g_0U_T\; ,
    $$
    $$
    V_T=U_T\cup g_0U_S\; ,
    $$
and
    $$
    V_P=U_P\cup g_0U_P\; .
    $$
 Then $V_O$ is a (not normal) subgroup in $G$ of index $4$. 
 The sets $V_O$, $V_S$, $V_T$, $V_P$ are left cosets of the 
 group $G$ by $V_O$. The cocycle $\phi _{O,\psi }$
 can be computed by the formula
    $$
    \phi _{O,\psi }(g)=\left\{
    \begin{array}{llll}
    O\; ,\; \, \hbox{if}\; \, g\in V_O \\
    S\; ,\; \, \hbox{if}\; \, g\in V_S \\
    T\; ,\; \, \hbox{if}\; \, g\in V_T \\
    P\; ,\; \, \hbox{if}\; \, g\in V_P
    \end{array}
    \right.
    $$
    \end{proposition}

\begin{pf}
We know that $G/U_O\cong D_4$. Then this proposition can be proved
by direct computation with using the formula
    \begin{equation}
    \label{kuznec}
    \phi _{O,\psi }(g)=\left\{
    \begin{array}{ll}
    \psi (g)\, , & \mbox{�᫨ $g\in H$} \\
    g_0\psi (g_0^{-1}g)\, , &
    \mbox{�᫨ $g\in G\backslash H\; .$}
    \end{array}
    \right.
    \end{equation}
and rules of multiplication in $G/U_O\cong D_4$. For example, let
us prove that $V_O$ is a subgroup in $G$ and that it is not normal
in $G$. By definition of a cocycle of type $III$, we have that
$\overline {g_0}^2=1$ in $G/U_O$. Hence $V_O=U_O\cup g_0U_O$ is a
subgroup in $G$. Let $h_S\in U_S$, $h_T\in U_T$ and $h_P\in U_P$.
Then for any $u\in U_O$ we have:
   $$
   \overline {h_S^{-1}g_0uh_S}=
   \overline {h_S}\overline {g_0}\overline {h_S}=
   \overline {g_0}
   (\overline {g_0}^{-1}\overline {h_S}\overline {g_0})
   \overline {h_S}=
   \overline {g_0}\overline {h_T}\overline {h_S}=
   \overline {g_0}\overline {h_P}=
   \overline {g_0h_P}\; .
   $$
In other words, $h_S^{-1}g_0uh_S\in g_0U_P$. Hence
$g_0^{-1}V_Og_0\not \subset V_O$ (because by definition of the set
$V_O$ we have that $V_O\cap g_0U_P=\emptyset $).
\end{pf}

We can illustrate the values of cocycle $\phi _{O,\psi }$ of type
$III$ by the diagram

\begin{center}

   \begin{tabular}[|c|c|]{|p{10mm}|p{10mm}|}
   \hline
   & \\
   $S$ & $P$ \\
   & \\
   \hline
   & \\
   $O$ & $T$ \\
   & \\
   \hline
   \hline
   & \\
   $T$ & $P$ \\
   & \\
   \hline
   & \\
   $O$ & $S$ \\
   & \\
   \hline
   \end{tabular}

\end{center}

\vspace{4mm}

\subsection{\it $\delta $ on cocycles}
\label{delta}

Now we want to lift the cohomology classes of cocycles $\phi
_{Q,\psi }$ to the group $_2\Br (E)$. In other words, we want to
describe unramified central simple algebras of an exponent $2$
such that its classes maps into classes $\cls (\phi _{Q,\psi })\in
H^1(G,\bar E)$ under the homomorphism $\pi $.

According to Section \ref{lebeda} and formula (\ref{tiski}) the
first step must be computing $\delta (\cls (\phi
_{Q,\psi }))$ for cycles of the $I$ and $II$ types. We will determine
$\delta (\cls (\phi _{Q,\psi }))$ in terms of explicit $2$-cocycles
of the group $G$ with coefficients in $\Pri (\bar E)$.

Consider the following particular case of the short exact sequence
(\ref{ji}):
  $$
  0\ra \Pri(\bar E)\lra \Div^0(\bar E) \stackrel{\rm{sum}}{\lra}
  \bar E\ra 0\; .
  $$
For any point $W\in \bar E$ denote by $(W)$ the same point viewed
as a divisor on $E$. Then
   $$
   \rm {sum }((W)-(O))=W\; .
   $$
Let $\phi $ be a continuous (in sense of the profinite topology on
$G$) $1$-cocycle of the group $G$ with coefficients in $\bar E$.
Denote by $\hat \phi $ the continuous map
    $$
    \hat {\phi }:G\lra \Div ^0(\bar E)
    $$
such that
    $$
    \hat {\phi }(g)=(\phi (g))-(O)
    $$
for any $g\in G$. Then $\pp \hat \phi $ is a $2$-cocycle of $G$ with
coefficients in $\Pri (\bar E)$, and
    $$
    \delta (\cls (\phi ))=\cls (\pp \hat \phi )\; .
    $$
The formula
    \begin{equation}
    \label{pp}
    \pp \hat {\phi }(g_1,g_2)=g_1\hat {\phi }(g_2)-
    \hat {\phi }(g_1g_2)+\hat {\phi }(g_2)
    \end{equation}
allows to compute $\pp \hat {\phi }$ by $\hat {\phi }$. We need
to do this for cocycles of types $I$, $II$ and $III$. First we
consider the cases of cocycles of $I$ and $II$ types.

\section{Quaternion and cyclic algebras}
\label{kamchatka}

\subsection{\it Algebras corresponding to cocyles of type $I$}
\label{chukotka}

   \begin{proposition}
   \label{mcensk}
   Let $\phi _{O,\psi }$ be a cocycle of type $I$,
   $U=\ker (\psi )$ and $W=U\cup g_0U$ the subgroup of index $2$ in
   $G$ (see Proposition \ref{vologda}). Consider an element
   $a\in k\backslash k^2$ such that $k(\sqrt {a})=\bar k^W$
   is the subfield invariant under the action of $W$.
   Then the quaternion algebra
      $$
      (a,x-x_P)
      $$
   with center $k(E)$ is unramified. The class of $(a,x-x_P)$ 
   maps into the class $\cls (\phi _{O,\psi })$ under the
   homomorphism $\pi :\Br (E)\ra H^1(G,\bar E)$.
   \end{proposition}
\begin{pf}
Consider the $2$-cocycle $\Phi _{P,a}:G\times G\ra \bar k(\bar E)^*$
defined by the formula
    $$
    \Phi _{P,a}(g_1,g_2)=\left\{
    \begin{array}{ll}
    x-x_P\; ,\; \, \hbox{if}\; \,
    (g_1,g_2)\in (G\backslash W)\times (G\backslash W) \\
    1\; ,\; \, \hbox{in the other case}
    \end{array}
    \right.
    $$
The cocycle $\Phi _{P,a}$ defines the quaternion algebra $(a,x-x_P)$ 
over $k(E)$. If $\di :\bar k(\bar E)^*\ra \Pri (\bar E)$ is the map 
sending $f\in \bar k(\bar E)^*$ into the divisor $\di (f)$ of zeros and 
poles of the function $f$, then $\pp \hat {\phi }_{O,\psi }=\di \circ 
\Phi _{P,a}$. Taking into account the diagram in Section \ref{lebeda}, 
one can see that
  $$
  \inv (\cls (\Phi _{P,a}))=
  \beta (\alpha (\cls (\Phi _{P,a})))=
  \beta (\cls (\di \circ \Phi _{P,a}))=
  \beta (\cls (\pp \hat {\phi }_{O,\psi }))=
  \beta (\delta (\cls ({\phi }_{O,\psi }))=0\; .
  $$
Hence the algebra $(a,x-x_P)$ is unramified and $\pi (\cls (\Phi
_{P,a}))=\cls (\phi _{O,\psi })$.
\end{pf}

\subsection{\it $\delta $ for cocycles of type $II$}
\label{rybinsk}

   \begin{proposition}
   \label{tver'}
   Let $\phi _{T,\psi }$ be a cocycle of type $II$.
   Then the $2$-cocycle $\pp \hat {\phi }_{T,\psi }:
   G\times G\ra \Pri (\bar E)$
   is given in the following table

\vspace{4mm}

\begin{center}

   {\scriptsize
   \begin{tabular}{|c|c|c|c|c|}
   \hline
   & &  &  & \\
   $g_2\in G\backslash (H\cup g_0U)$ &
   $0$ & $(S)-(T)+(P)-(O)$ & $2(T)-2(O)$ & $(T)-(P)+(S)-(O)$ \\
   & &  &  & \\
   \hline & &  &  & \\
   $g_2\in g_0U$ &
   $0$ & $(T)-(S)+(P)-(O)$ & $(S)-(P)+(T)-(O)$ & $2(S)-2(O)$ \\
   & &  &  & \\
   \hline
   & &  &  & \\
   $g_2\in H\backslash U$ &
   $0$ & $2(P)-2(O)$ & $(P)-(S)+(T)-(O)$ & $(P)-(T)+(S)-(O)$ \\
   & &  &  & \\
   \hline
   & &  &  & \\
   $g_2\in U$ &
   $0$ & $0$ & $0$ & $0$ \\
   & &  &  & \\
   \hline
   & &  &  & \\
   & $g_1\in U$ & $g_1\in H\backslash U$ & $g_1\in g_0U$ & $g_1\in
   G\backslash (H\cup g_0U)$ \\
   & &  &  & \\
   \hline
   \end{tabular}
   }

\end{center}

   \end{proposition}
\begin{pf}
By Proposition \ref{vologda} we have the following explicit formula
for $\hat {\phi }_{T,\psi }$:

    \begin{equation}
    \label{saratov}
    \hat {\phi }_{T,\psi }(g)=\left\{
    \begin{array}{llll}
    0\; ,\; \, \; \, g\in U \Lra i=0 \\
    (T)-(O)\; ,\; \, \; \, g\in g_0U \Lra i=1 \\
    (P)-(O)\; ,\; \, \; \, g\in H\backslash U \Lra i=2 \\
    (S)-(O)\; ,\; \, \; \, g\in G\backslash (H\cup g_0U) \Lra
    i=3
    \end{array}
    \right.
    \end{equation}

Let $g_1,g_2\in G$ and $i_1$, $i_2$ are the orders of $g_1$, $g_2$
respectively. It is clear that the order $i$ of $g_1g_2$ is equal
to $i_1+i_2$ modulo $4$. The direct computation with the formulas
(\ref{saratov}) and (\ref{pp}) gives the following explicit form of the
cocycle $\pp \hat {\phi }_{T,\psi }$:

\vspace{4mm}

\begin{center}

  \begin{tabular}{|c|c|c|c|c|c|}
  \hline $i_1$ & $i_2$ & $g_1\phi _{T,\psi }(g_2)$ &
  $-\phi _{T,\psi }(g_1g_2)$ & $\phi _{T,\psi }(g_1)$ & sum \\
  \hline
  \hline
  $0$ & $0$ & $0$ & $0$ & $0$ & $0$ \\
  \hline
  $0$ & $1$ & $(T)-(O)$ & $-(T)+(O)$ & $0$ & $0$ \\
  \hline
  $0$ & $2$ & $(P)-(O)$ & $-(P)+(O)$ & $0$ & $0$ \\
  \hline
  $0$ & $3$ & $(S)-(O)$ & $-(S)+(O)$ & $0$ & $0$ \\
  \hline
  \hline
  $1$ & $0$ & $0$ & $-(T)+(O)$ & $(T)-(O)$ & $0$ \\
  \hline
  $1$ & $1$ & $(S)-(O)$ & $-(P)+(O)$ & $(T)-(O)$ & $(S)-(P)+(T)-(O)$ \\
  \hline
  $1$ & $2$ & $(P)-(O)$ & $-(S)+(O)$ & $(T)-(O)$ & $(P)-(S)+(T)-(O)$ \\
  \hline
  $1$ & $3$ & $(T)-(O)$ & $0$ & $(T)-(O)$ & $2(T)-2(O)$ \\
  \hline
  \hline
  $2$ & $0$ & $0$ & $-(P)+(O)$ & $(P)-(O)$ & $0$ \\
  \hline
  $2$ & $1$ & $(T)-(O)$ & $-(S)+(O)$ & $(P)-(O)$ & $(T)-(S)+(P)-(O)$ \\
  \hline
  $2$ & $2$ & $(P)-(O)$ & $0$ & $(P)-(O)$ & $2(P)-2(O)$ \\
  \hline
  $2$ & $3$ & $(S)-(O)$ & $-(T)+(O)$ & $(P)-(O)$ & $(S)-(T)+(P)-(O)$ \\
  \hline
  \hline
  $3$ & $0$ & $0$ & $-(S)+(O)$ & $(S)-(O)$ & $0$ \\
  \hline
  $3$ & $1$ & $(S)-(O)$ & $0$ & $(S)-(O)$ & $2(S)-2(O)$ \\
  \hline
  $3$ & $2$ & $(P)-(O)$ & $-(T)+(O)$ & $(S)-(O)$ & $(P)-(T)+(S)-(O)$ \\
  \hline
  $3$ & $3$ & $(T)-(O)$ & $-(P)+(O)$ & $(S)-(O)$ & $(T)-(P)+(S)-(O)$ \\
  \hline
  \end{tabular}

\end{center}

\vspace{4mm}

\noindent Rewriting this table in more compact form we obtain the
statement of the proposition.
\end{pf}

\subsection{\it Cross products corresponding to cocycles of type $II$}
\label{mars}

We need some lemma. For short we set
   $$
   D_P=2(P)-2(O)=\di (x-x_P)\; ,
   $$
   $$
   D_T=2(T)-2(O)=\di (x-x_T)\; ,
   $$
   $$
   D_S=2(T)-2(O)=\di (x-x_S)\; ,
   $$
and
   $$
   D=(P)+(T)+(S)-3(O)\; .
   $$
Let
   $$
   \Gamma =\langle D,D_P,D_T,D_S\rangle
   $$
be the subgroup in $\Pri (\bar E)$ generated by these principle
divisors. Also let
   $$
   \Omega =\langle y,x-x_P,x-x_T,x-x_S\rangle
   $$
be the subgroup in $\bar k(\bar E)^*$ generated by the functions
$x-x_P$, $x-x_T$, $x-x_S$ and $y$. It is clear that $\Gamma $ and
$\Omega $ are discrete $G$-submodules in $\Pri (\bar E)$ and
$\bar k(\bar E)^*$ respectively. The restriction of the
homomorphism $\di :\bar k(\bar E)^*\ra \Pri (\bar E)$ on the submodule
$\Gamma $ is a surjective $G$-homomorphism
    $$
    \xymatrix{\Omega \ar@{>>}[r]^-{\di } & \Gamma }\; .
    $$
   \begin{lemma}
   \label{tambov}
   Let $F=F\langle D,D_P,D_T,D_S\rangle $ denotes the free abelian group
   generated by the set $\{ D,D_P,D_T,D_S\} $. Let $\tau :F\ra \Omega $
   be the homomorphism defined by $\tau(D_P)=x-x_P$, $\tau (D_T)=x-x_T$,
   $\tau (D_S)=x-x_S$, $\tau (D)=y$. Let $\mu :F\ra \Gamma $
   be the homomorphism which sends any element of the basis of the group $F$
   to itself. Then there exists a unique homomorphism
   (of discrete $G$-modules) $\gamma $, which makes the diagram
   $$
   \xymatrix{ F \ar[r]^-{\mu } \ar[dr]_-{\tau } &
   \Gamma \ar[d]^-{\gamma } \\
   & \Omega }
   $$
   commutative. In particular,
   $$
   \di \circ \gamma ={\rm id}_{\Gamma }\; .
   $$
   \end{lemma}
\begin{pf}
Observe that $2D=D_P+D_T+D_S$. Show that it is a unique relation in
$\Gamma $. Indeed, let $l,m,n,t\in \bZ $ and $D+mD_P+nD_T+tD_S=0$.
Then
   $$
   l(P)+l(T)+l(S)-3l(O)+
   2m(P)-2m(O)+
   2n(T)-2n(O)+
   2t(S)-2t(O)=
   $$
   $$
   (l+2m)(P)+(l+2n)(T)+(l+2t)(S)-2(m+n+t)(O)-3l(O)=0 \Ra
   $$
   $$
   \Ra
   \left\{
   \begin{array}{llll}
   l+2m=0 \\
   l+2n=0 \\
   l+2t=0 \\
   2(m+n+t)+3l=0
   \end{array}
   \right.
   $$
Solving this system one can see that $m=n=t$ and $l=-2m$. Hence the
relation $lD+mD_P+nD_T+tD_S=0$ is of the form
$-2mD+mD_P+mD_T+mD_S=0$, that is $2D=D_P+D_T+D_S$.

Thus we have that $\Gamma $ is the quotient group of $F=F\langle
D,D_P,D_T,D_S\rangle $ by the cyclic subgroup generated by the
element $D_P+D_T+D_S-2D$. Then the existence of $\gamma $ follows
from the equality
   $$
   \tau
   (D_P+D_T+D_S-2D)=\frac{(x-x_P)(x-x_T)(x-x_S)}{y^2}=
   \frac{y^2}{y^2}=1\; .
   $$
\end{pf}

   \begin{definition}
   \label{orel}
   Let $\phi _{T,\psi }$ be a cocycle of type $II$ defined by
   a normal subgroup $U$ in $G$ such that $U\subset H$ and
   $G/U\cong \bZ /4$. Let
   $$
   \Phi _{T,\psi }:=\gamma \circ \pp \hat {\phi }_{T,\psi }\; .
   $$
   Since $\gamma $ is a homomorphism of discrete $G$-modules
   and $\pp \hat {\phi }_{T,\psi }$ is a $2$-cocycle, the map
   $\Phi _{T,\psi }$ is a $2$-cocycle of the group $G$ with coefficients
   in $\bar k(\bar E)^*$. Let
   $$
   A_U
   $$
   be the central simple algebra over $k(E)$ which is the cross product
   defined by the cocycle $\Phi _{T,\psi }$.
   \end{definition}

  \begin{proposition}
  \label{sochi}
  Let $\phi _{T,\psi }$ be a cocycle of type $II$ and
  $U=\ker (\psi )$. Then the $2$-cocyle $\Phi _{T,\psi }$ from Definition
  \ref{orel} can be explicitly given by the following table

\vspace{4mm}

\begin{center}

   \begin{tabular}{|c|c|c|c|c|}
   \hline
   & &  &  & \\
   $g_2\in G\backslash (H\cup g_0U)$ &
   $1$ & $\frac{y}{x-x_T}$ & $x-x_T$ & $\frac{y}{x-x_P}$ \\
   & &  &  & \\
   \hline & &  &  & \\
   $g_2\in g_0U$ &
   $1$ & $\frac{y}{x-x_S}$ & $\frac{y}{x-x_P}$ & $x-x_S$ \\
   & &  &  & \\
   \hline
   & &  &  & \\
   $g_2\in H\backslash U$ &
   $1$ & $x-x_P$ & $\frac{y}{x-x_S}$ & $\frac{y}{x-x_T}$ \\
   & &  &  & \\
   \hline
   & &  &  & \\
   $g_2\in U$ &
   $1$ & $1$ & $1$ & $1$ \\
   & &  &  & \\
   \hline
   & &  &  & \\
   & $g_1\in U$ & $g_1\in H\backslash U$ & $g_1\in g_0U$ & $g_1\in
   G\backslash (H\cup g_0U)$ \\
   & &  &  & \\
   \hline
   \end{tabular}

\end{center}

\vspace{4mm}

  The cross product $A_U$ is an unramified central simple
  algebra of exponent $2$ over the function field $k(E)$.
  The class of $A_U$ in $\Br (E)$ maps into the class
  $\cls (\phi _{T,\psi })$ under the homomorphism $\pi :\Br (E)\ra
  H^1(G,\bar E)$.
  \end{proposition}
\begin{pf}
To obtain the table it is sufficient to apply the homomorphism
$\gamma $ to the table \ref{tver'}. Since $\di \circ \gamma ={\rm
id}_{\Gamma }$, it follows that $\di \circ \Phi _{T,\psi }=\di \circ \gamma
\circ \pp \hat {\phi }_{T,\psi }=\pp \hat {\phi }_{T,\psi }$.
Hence $\alpha (\cls (\Phi _{T,\psi }))=\cls (\pp \hat {\phi
}_{T,\psi })=\delta (\cls (\phi _{T,\psi }))$. Taking into account
the commutative diagram from Section \ref{lebeda}, we see that
   $$
   \inv (\cls (\Phi _{T,\psi }))=
   \beta \circ \alpha (\cls (\Phi _{T,\psi }))=
   \beta \circ \delta (\cls (\phi _{T,\psi }))=0\; .
   $$
Therefore, $\cls (\Phi _{T,\psi })\in \ker (\inv )$. This means
that $A_U$ is an unramified algebra over $k(E)$. The exponent of $A_U$
is equal to $2$ because $2\phi _{T,\psi }=0$. The last assertion of
the proposition is a direct consequence of the equality $\delta
(\cls (\phi _{T,\psi }))=\alpha (\cls (\Phi _{T,\psi }))$ and
the definition of the homomorphism $\pi $.
\end{pf}

\subsection{\it Cyclic algebras of degree $4$}
\label{four}

Note that the algebra $A_U$ splits by a cyclic extension of degree
four. To find a corresponding cyclic algebra we need some lemma
relating to second cohomology of a cyclic group of order
$4$. Let $C=\{ 1,c,c^2,c^4\} $ be a such group with a generator $c$.
Let $M$ be a $C$-module, $M^C=\{ m\in M\palka cm=m\} $ and $N(M)=\{
m+cm\palka m\in M\} $. Then we have the well known isomorphism
   $$
   v:H^2(C,M)\lra M^C/N(M)
   $$
(see, for example, \cite{Mac}, chapter IV, $\S $ 7).
   \begin{lemma}
   \label{kerch'}
   Let $\phi :C\times C\ra M$ be a $2$-cocycle of $C$ with
   coefficients in $M$. Then
   $$
   v(\cls (\phi ))=\phi (c^3,c)+\phi (1,1)+\phi (c^2,c)+\phi (c,c)+N(A)\; .
   $$
   \end{lemma}
\begin{pf}
The proof can be provided by comparison of two standard
resolutions of the trivial $C$-module $\bZ $ for the cyclic group
$C=\{ 1,c,c^2,c^3\} $.
\end{pf}

We use Lemma \ref{kerch'} for computation of cyclic algebras of
degree $4$ similar to cross products $A_U$.

   \begin{proposition}
   \label{suzdol'}
   Let $\phi _{T,\psi }$ be a cocycle of type II,
   $U=\ker (\psi )$, $A_U$ be the central simple algebra
   defined by $\Phi _{T,\psi }$ (see Definition \ref{orel}).
   Let $Z/k$ be the cyclic field extension of degree $4$ defined by
   $U$. Then the algebra $A_U$ is similar to the cyclic algebra
       $$
       (Z\cdot k(E)/k(E),f)
       $$
   over center $k(E)$. Here $f=f(x)$ is the polynomial
   from the decomposition (\ref{fom}) considered as an element in $k(E)$.
   \end{proposition}
\begin{pf}
Let $C=G/U$ be the cyclic group of order $4$ with the generator
$c=g_0U$, $Z={\bar k}^C$ and $Z(E)$ be the field of $Z$-rational
functions on the curve $E_Z=E\times \Spec (Z)$. The table from
Proposition \ref{sochi} shows that the $2$-cocycle $\Phi _{T,\psi }$
factors through the group $C$, that is there exists the
commutative square
    $$
    \xymatrix{
    G\times G \ar[rr]^-{} \ar[dd]_-{\Phi _{T,\psi }} & &
    C\times C \ar[dd]^-{\varphi _{T,\psi }} \\ \\
    \bar k(\bar E)^* & & Z(E)^* \ar[ll]
    }
    $$
where the cocyle $\varphi _{T,\psi }:C\times C\ra Z(E)^*$ can be
computed according to the table

\vspace{4mm}

\begin{center}

   \begin{tabular}{|c|c|c|c|c|}
   \hline
   & &  &  & \\
   $c^3$ &
   $1$ & $\frac{y}{x-x_T}$ & $x-x_T$ & $\frac{y}{x-x_P}$ \\
   & &  &  & \\
   \hline & &  &  & \\
   $c$ &
   $1$ & $\frac{y}{x-x_S}$ & $\frac{y}{x-x_P}$ & $x-x_S$ \\
   & &  &  & \\
   \hline
   & &  &  & \\
   $c^2$ &
   $1$ & $x-x_P$ & $\frac{y}{x-x_S}$ & $\frac{y}{x-x_T}$ \\
   & &  &  & \\
   \hline
   & &  &  & \\
   $1$ &
   $1$ & $1$ & $1$ & $1$ \\
   & &  &  & \\
   \hline
   & &  &  & \\
   & $1$ & $c^2$ & $c$ & $c^3$ \\
   & &  &  & \\
   \hline
   \end{tabular}

\end{center}

\vspace{4mm}

\noindent The cohomology class $\cls (\Phi _{T,\psi })\in
H^2(G,\bar k(\bar E)^*)$ is the image of $\cls (\varphi _{T,\psi
})\in H^2(C,Z(E)^*)$ under the injective inflation homomorphism $
\infl :H^2(C,Z(E)^*)\hra H^2(G,\bar k(\bar E)^*)$. In terms of
central simple algebras this means that the algebra $A_U\otimes
_{k(E)}Z(E)$ splits, so the class of algebra $A_U$ lies in
the relative Brauer group $\Br (Z(E)/k(E))$.

Now observe that for $M=Z(E)^*$ the norm homomorphism
$N:Z(E)^*\ra Z(E)^*$ is the norm $N_{Z(E)/k(E)}:Z(E)^*\ra k(E)^*$
in the sense of field extensions. According to Lemma \ref{kerch'}
the isomorphism
   $$
   v:H^2(C,Z(E)^*)\stackrel{\cong }{\lra }k(E)^*/N(Z(E)^*)
   $$
sends $\varphi _{T,\psi }$ into the class of the rational function
   $$
   \varphi _{T,\psi }(c^3,c)\cdot
   \varphi _{T,\psi }(1,1)\cdot
   \varphi _{T,\psi }(c^2,c)\cdot
   \varphi _{T,\psi }(c,c)=
   $$
   $$
   =(x-x_S)\cdot \frac{y}{x-x_S}\cdot \frac{y}{x-x_P}=
   $$
   $$
   =\frac{y^2}{x-x_P}=\frac{(x-x_P)\cdot f(x)}{x-x_P}=f(x)\; .
   $$

Let $(Z\cdot k(E)/k(E),f(x))$ be the cyclic algebra over $k(E)$ defined by the
cyclic extension $Z\cdot k(E)/k(E)$, the generator $c=g_0U$ of the group 
$C=G/U$ and the rational function $f(x)\in k(E)$. Then $(Z\cdot k(E)/k(E),f(x))$ 
is the crossed product induced by the $2$-cocycle

\vspace{4mm}

\begin{center}

   \begin{tabular}{|c|c|c|c|c|}
   \hline
   $c^3$ &
   $1$ & $f$ & $f$ & $f$ \\
   \hline
   $c$ &
   $1$ & $1$ & $1$ & $f$ \\
   \hline
   $c^2$ &
   $1$ & $f$ & $1$ & $f$ \\
   \hline
   $1$ &
   $1$ & $1$ & $1$ & $1$ \\
   \hline
   & $1$ & $c^2$ & $c$ & $c^3$ \\
   \hline
   \end{tabular}

\end{center}

\vspace{4mm}

Using Lemma (\ref{kerch'}) once more we see that the isomorphism
$$v:H^2(C,Z(E)^*)\stackrel{\cong }{\ra }k(E)^*/N(Z(E)^*)$$ maps the
class of $\varphi '_{T,\psi }$ into the coset of the rational
function
   $$
   \varphi '_{T,\psi }(c^3,c)\cdot
   \varphi '_{T,\psi }(1,1)\cdot
   \varphi '_{T,\psi }(c^2,c)\cdot
   \varphi '_{T,\psi }(c,c)=
   f\cdot 1\cdot 1\cdot 1=f\; .
   $$
Thus we have that $v(\cls (\phi '_{T,\psi }))=v(\cls (\phi
_{T,\psi }))$, whence $\cls (\phi '_{T,\psi })=\cls (\phi _{T,\psi
})$. In terms of central simple algebras over $k(E)$ this means
that the algebra $A_U$ is similar to the algebra $(Z\cdot k(E)/k(E),f)$.
\end{pf}

\section{Dihedral algebras}
\label{dihedral}

Now let us calculate unramified central simple algebras over
$k(E)$ corresponding to cocycles $\phi _{O,\psi }:G\ra \dv
\bar E$ of type $III$. Cocycles of type $III$ was defined in
Section \ref{kwazar} and calculated in Section \ref{pluton}. {\it
We shall use the notations of these sections.} In particular, $U$
is a subgroup in $H$ such that $[H:U]=2$ and $U\neq
g_0^{-1}Ug_0$. If $\psi _T:H\ra \{ O,T\} $ is the surjective
homomorphism with kernel $U$ and $\psi _S:H\ra \{ O,T\} $ is the
surjective homomorphism with kernel $g_0^{-1}Ug_0$, then $\psi
=\psi _T+\psi _S$.

\subsection{\it $\delta $ for cocycles of type $III$}
\label{arhimed}

  \begin{proposition}
  \label{upiter}
  Let $\phi _{O,\psi }$ be a cocycle of type $III$ defined by
  a subgroup $U$ of index $2$ in $H$ such that $g_0^{-1}Ug_0\neq
  U$, where $g_0^2\in U_O=U\cap g_0^{-1}Ug_0$ (see Sections
  \ref{kwazar} and \ref{pluton}).
  Let $\hat \phi _{O,\psi }:G\ra \Div ^0(\bar E)$
  be the map such that $\hat \phi _{O,\psi }(g)=(Q)-(O)$ if and only
  if $\phi _{O,\psi }(g)=Q\in \dv \bar E$. Then the $2$-cocycle
  $\pp \hat \phi _{O,\psi }:G\times G\ra \Pri (\bar E)$
  can be calculated by the formula
  $$
  \phi _{O,\psi }(g_1,g_2)=\left\{
  \begin{array}{llll}
  0\; ,\; \;
  (g_1,g_2)\in (V_O\times G)\cup (G\times V_O) \\
  \di (x-x_S)\; ,\; \;
  (g_1,g_2)\in (U_S\times V_S)\cup (g_0U_T\times V_T) \\
  \di (x-x_T)\; ,\; \;
  (g_1,g_2)\in (U_T\times V_T)\cup (g_0U_S\times V_S) \\
  \di (x-x_P)\; ,\; \;
  (g_1,g_2)\in V_P\times V_P \\
  \di (\frac{y}{x-x_S})\; ,\; \;
  (g_1,g_2)\in (V_T\times V_P)\cup (U_P\times V_T)\cup (g_0U_P\times V_S) \\
  \di (\frac{y}{x-x_T})\; ,\; \;
  (g_1,g_2)\in (V_S\times V_P)\cup (U_P\times V_S)\cup (g_0U_P\times V_T) \\
  \di (\frac{y}{x-x_P})\; ,\; \;
  (g_1,g_2)\in ((U_S\cup g_0U_S)\times V_T)\cup ((U_T\cup g_0U_T)\times V_S) \\
  \end{array}
  \right.
  $$
  \end{proposition}
\begin{pf}
For the proof we use the formula
   $$
   \pp \hat \phi _{O,\psi }(g_1,g_2)=g_1\hat \phi _{O,\psi }(g_2)-
   \hat \phi _{O,\psi }(g_1g_2)+\hat \phi _{O,\psi }(g_1)
   $$
and the rule of multiplication in $G/U_O\cong D_4$. All calculations
are in the following eight tables:

%%%%%% tablica 1 + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{4mm}

\begin{center}

 \begin{tabular}{|c|c||c|c|c||c|}
 \hline $g_1$ & $g_2$ & $g_1\phi _{O,\psi }(g_2)$ &
 $-\phi _{O,\psi }(g_1g_2)$ & $\phi _{O,\psi }(g_1)$ & sum \\
 \hline
 \hline
 $U_O$ & $U_O$    & $0$ & $0$ & $0$ & $0$ \\
 \hline
 $U_O$ & $U_S$    & $(S)-(O)$ & $-(S)+(O)$ & $0$ & $0$ \\
 \hline
 $U_O$ & $U_T$    & $(T)-(O)$ & $-(T)+(O)$ & $0$ & $0$ \\
 \hline
 $U_O$ & $U_P$    & $(P)-(O)$ & $-(P)+(O)$ & $0$ & $0$ \\
 \hline
 \hline
 $U_O$ & $g_0U_O$ & $0$ & $0$ & $0$ & $0$ \\
 \hline
 $U_O$ & $g_0U_S$ & $(T)-(O)$ & $-(T)+(O)$ & $0$ & $0$ \\
 \hline
 $U_O$ & $g_0U_T$ & $(S)-(O)$ & $-(S)+(O)$ & $0$ & $0$ \\
 \hline
 $U_O$ & $g_0U_P$ & $(P)-(O)$ & $-(P)+(O)$ & $0$ & $0$ \\
 \hline
 \end{tabular}

%%%%%% tablica 2 + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{4mm}

 \begin{tabular}{|c|c||c|c|c||c|}
 \hline $g_1$ & $g_2$ & $g_1\phi _{O,\psi }(g_2)$ &
 $-\phi _{O,\psi }(g_1g_2)$ & $\phi _{O,\psi }(g_1)$ & sum \\
 \hline
 \hline
 $U_S$ & $U_O$ & $0$ & $-(S)+(O)$ & $(S)-(O)$ & $0$ \\
 \hline
 $U_S$ & $U_S$ & $(S)-(O)$ & $0$ & $(S)-(O)$ & $2(S)-2(O)$ \\
 \hline
 $U_S$ & $U_T$ & $(T)-(O)$ & $-(P)+(O)$ & $(S)-(O)$ & $(T)-(P)+(S)-(O)$ \\
 \hline
 $U_S$ & $U_P$ & $(P)-(O)$ & $-(T)+(O)$ & $(S)-(O)$ & $(P)-(T)+(S)-(O)$ \\
 \hline
 \hline
 $U_S$ & $g_0U_O$ & $0$ & $-(S)+(O)$ & $(S)-(O)$ & $0$ \\
 \hline
 $U_S$ & $g_0U_S$ & $(T)-(O)$ & $-(P)+(O)$ & $(S)-(O)$ & $(T)-(P)+(S)-(O)$ \\
 \hline
 $U_S$ & $g_0U_T$ & $(S)-(O)$ & $0$ & $(S)-(O)$ & $2(S)-2(O)$ \\
 \hline
 $U_S$ & $g_0U_P$ & $(P)-(O)$ & $-(T)+(O)$ & $(S)-(O)$ & $(P)-(T)+(S)-(O)$ \\
 \hline
 \end{tabular}

%%%%%% tablica 3 + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{4mm}

 \begin{tabular}{|c|c||c|c|c||c|}
 \hline $g_1$ & $g_2$ & $g_1\phi _{O,\psi }(g_2)$ &
 $-\phi _{O,\psi }(g_1g_2)$ & $\phi _{O,\psi }(g_1)$ & sum \\
 \hline
 \hline
 $U_T$ & $U_O$ & $0$ & $-(T)+(O)$ & $(T)-(O)$ & $0$ \\
 \hline
 $U_T$ & $U_S$ & $(S)-(O)$ & $-(P)+(O)$ & $(T)-(O)$ & $(S)-(P)+(T)-(O)$ \\
 \hline
 $U_T$ & $U_T$ & $(T)-(O)$ & $0$ & $(T)-(O)$ & $2(T)-2(O)$ \\
 \hline
 $U_T$ & $U_P$ & $(P)-(O)$ & $-(S)+(O)$ & $(T)-(O)$ & $(P)-(S)+(T)-(O)$ \\
 \hline
 \hline
 $U_T$ & $g_0U_O$ & $0$ & $-(T)+(O)$ & $(T)-(O)$ & $0$ \\
 \hline
 $U_T$ & $g_0U_S$ & $(T)-(O)$ & $0$ & $(T)-(O)$ & $2(T)-2(O)$ \\
 \hline
 $U_T$ & $g_0U_T$ & $(S)-(O)$ & $-(P)+(O)$ & $(T)-(O)$ & $(S)-(P)+(T)-(O)$ \\
 \hline
 $U_T$ & $g_0U_P$ & $(P)-(O)$ & $-(S)+(O)$ & $(T)-(O)$ & $(P)-(S)+(T)-(O)$ \\
 \hline
 \end{tabular}

%%%%%% tablica 4 + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{4mm}

 \begin{tabular}{|c|c||c|c|c||c|}
 \hline $g_1$ & $g_2$ & $g_1\phi _{O,\psi }(g_2)$ &
 $-\phi _{O,\psi }(g_1g_2)$ & $\phi _{O,\psi }(g_1)$ & sum \\
 \hline
 \hline
 $U_P$ & $U_O$ & $0$ & $-(P)+(O)$ & $(P)-(O)$ & $0$ \\
 \hline
 $U_P$ & $U_S$ & $(S)-(O)$ & $-(T)+(O)$ & $(P)-(O)$ & $(S)-(T)+(P)-(O)$ \\
 \hline
 $U_P$ & $U_T$ & $(T)-(O)$ & $-(S)+(O)$ & $(P)-(O)$ & $(T)-(S)+(P)-(O)$ \\
 \hline
 $U_P$ & $U_P$ & $(P)-(O)$ & $0$ & $(P)-(O)$ & $2(P)-2(O)$ \\
 \hline
 \hline
 $U_P$ & $g_0U_O$ & $0$ & $-(P)+(O)$ & $(P)-(O)$ & $0$ \\
 \hline
 $U_P$ & $g_0U_S$ & $(T)-(O)$ & $-(S)+(O)$ & $(P)-(O)$ & $(T)-(S)+(P)-(O)$ \\
 \hline
 $U_P$ & $g_0U_T$ & $(S)-(O)$ & $-(T)+(O)$ & $(P)-(O)$ & $(S)-(T)+(P)-(O)$ \\
 \hline
 $U_P$ & $g_0U_P$ & $(P)-(O)$ & $0$ & $(P)-(O)$ & $2(P)-2(O)$ \\
 \hline
 \end{tabular}

%%%%%% tablica 5 + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{4mm}

 \begin{tabular}{|c|c||c|c|c||c|}
 \hline $g_1$ & $g_2$ & $g_1\phi _{O,\psi }(g_2)$ &
 $-\phi _{O,\psi }(g_1g_2)$ & $\phi _{O,\psi }(g_1)$ & sum \\
 \hline
 \hline
 $g_0U_O$ & $U_O$    & $0$ & $0$ & $0$ & $0$ \\
 \hline
 $g_0U_O$ & $U_S$    & $(T)-(O)$ & $-(T)+(O)$ & $0$ & $0$ \\
 \hline
 $g_0U_O$ & $U_T$    & $(S)-(O)$ & $-(S)+(O)$ & $0$ & $0$ \\
 \hline
 $g_0U_O$ & $U_P$    & $(P)-(O)$ & $-(P)+(O)$ & $0$ & $0$ \\
 \hline
 \hline
 $g_0U_O$ & $g_0U_O$ & $0$ & $0$ & $0$ & $0$ \\
 \hline
 $g_0U_O$ & $g_0U_S$ & $(S)-(O)$ & $-(S)+(O)$ & $0$ & $0$ \\
 \hline
 $g_0U_O$ & $g_0U_T$ & $(T)-(O)$ & $-(T)+(O)$ & $0$ & $0$ \\
 \hline
 $g_0U_O$ & $g_0U_P$ & $(P)-(O)$ & $-(P)+(O)$ & $0$ & $0$ \\
 \hline
 \end{tabular}

%%%%%% tablica 6 + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{4mm}

 \begin{tabular}{|c|c||c|c|c||c|}
 \hline $g_1$ & $g_2$ & $g_1\phi _{O,\psi }(g_2)$ &
 $-\phi _{O,\psi }(g_1g_2)$ & $\phi _{O,\psi }(g_1)$ & sum \\
 \hline
 \hline
 $g_0U_S$ & $U_O$ & $0$ & $-(T)+(O)$ & $(T)-(O)$ & $0$ \\
 \hline
 $g_0U_S$ & $U_S$ & $(T)-(O)$ & $0$ & $(T)-(O)$ & $2(T)-2(O)$ \\
 \hline
 $g_0U_S$ & $U_T$ & $(S)-(O)$ & $-(P)+(O)$ & $(T)-(O)$ & $(S)-(P)+(T)-(O)$ \\
 \hline
 $g_0U_S$ & $U_P$ & $(P)-(O)$ & $-(S)+(O)$ & $(T)-(O)$ & $(P)-(S)+(T)-(O)$ \\
 \hline
 \hline
 $g_0U_S$ & $g_0U_O$ & $0$ & $-(T)+(O)$ & $(T)-(O)$ & $0$ \\
 \hline
 $g_0U_S$ & $g_0U_S$ & $(S)-(O)$ & $-(P)+(O)$ & $(T)-(O)$ & $(S)-(P)+(T)-(O)$ \\
 \hline
 $g_0U_S$ & $g_0U_T$ & $(T)-(O)$ & $0$ & $(T)-(O)$ & $2(T)-2(O)$ \\
 \hline
 $g_0U_S$ & $g_0U_P$ & $(P)-(O)$ & $-(S)+(O)$ & $(T)-(O)$ & $(P)-(S)+(T)-(O)$ \\
 \hline
 \end{tabular}

%%%%%% tablica 7 + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{4mm}

 \begin{tabular}{|c|c||c|c|c||c|}
 \hline $g_1$ & $g_2$ & $g_1\phi _{O,\psi }(g_2)$ &
 $-\phi _{O,\psi }(g_1g_2)$ & $\phi _{O,\psi }(g_1)$ & sum \\
 \hline
 \hline
 $g_0U_T$ & $U_O$ & $0$ & $-(S)+(O)$ & $(S)-(O)$ & $0$ \\
 \hline
 $g_0U_T$ & $U_S$ & $(T)-(O)$ & $-(P)+(O)$ & $(S)-(O)$ & $(T)-(P)+(S)-(O)$ \\
 \hline
 $g_0U_T$ & $U_T$ & $(S)-(O)$ & $0$ & $(S)-(O)$ & $2(S)-2(O)$ \\
 \hline
 $g_0U_T$ & $U_P$ & $(P)-(O)$ & $-(T)+(O)$ & $(S)-(O)$ & $(P)-(T)+(S)-(O)$ \\
 \hline
 \hline
 $g_0U_T$ & $g_0U_O$ & $0$ & $-(S)+(O)$ & $(S)-(O)$ & $0$ \\
 \hline
 $g_0U_T$ & $g_0U_S$ & $(S)-(O)$ & $0$ & $(S)-(O)$ & $2(S)-2(O)$ \\
 \hline
 $g_0U_T$ & $g_0U_T$ & $(T)-(O)$ & $-(P)+(O)$ & $(S)-(O)$ & $(T)-(P)+(S)-(O)$ \\
 \hline
 $g_0U_T$ & $g_0U_P$ & $(P)-(O)$ & $-(T)+(O)$ & $(S)-(O)$ & $(P)-(T)+(S)-(O)$ \\
 \hline
 \end{tabular}

%%%%%% tablica 8 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{4mm}

  \begin{tabular}{|c|c||c|c|c||c|}
  \hline $g_1$ & $g_2$ & $g_1\phi _{O,\psi }(g_2)$ &
  $-\phi _{O,\psi }(g_1g_2)$ & $\phi _{O,\psi }(g_1)$ & sum \\
  \hline
  \hline
  $g_0U_P$ & $U_O$    & $0$ & $-(P)+(O)$ & $(P)-(O)$ & $0$ \\
  \hline
  $g_0U_P$ & $U_S$    & $(T)-(O)$ & $-(S)+(O)$ & $(P)-(O)$ & $(T)-(S)+(P)-(O)$ \\
  \hline
  $g_0U_P$ & $U_T$    & $(S)-(O)$ & $-(T)+(O)$ & $(P)-(O)$ & $(S)-(T)+(P)-(O)$ \\
  \hline
  $g_0U_P$ & $U_P$    & $(P)-(O)$ & $0$ & $(P)-(O)$ & $2(P)-2(O)$ \\
  \hline
  \hline
  $g_0U_P$ & $g_0U_O$ & $0$ & $-(P)+(O)$ & $(P)-(O)$ & $0$ \\
  \hline
  $g_0U_P$ & $g_0U_S$ & $(S)-(O)$ & $-(T)+(O)$ & $(P)-(O)$ & $(S)-(T)+(P)-(O)$ \\
  \hline
  $g_0U_P$ & $g_0U_T$ & $(T)-(O)$ & $-(S)+(O)$ & $(P)-(O)$ & $(T)-(S)+(P)-(O)$ \\
  \hline
  $g_0U_P$ & $g_0U_P$ & $(P)-(O)$ & $0$ & $(P)-(O)$ & $2(P)-2(O)$ \\
  \hline
  \end{tabular}

\end{center}

\vspace{4mm}

Now one can rewrite the results of these calculations in more
compact way. This yields the third assertion of the proposition.
\end{pf}

\subsection{\it Dihedral cross products}
\label{meteor}

Now observe that the image of the cocycle $\pp \hat \phi _{O,\psi }$
lies in $\Gamma $ (see Section \ref{mars} for the definition of
$\Gamma $). Hence one can obtain the cocycle
   $$
   \Phi _{O,\psi }=\gamma \circ \pp \hat \phi _{O,\psi }:G\times
   G\lra \Omega \subset \bar k(\bar E)^*\; .
   $$
According Proposition \ref{upiter} the cocycle $\Phi _{O,\psi
}$ can be calculated by the formula
  $$
  \Phi _{O,\psi }(g_1,g_2)=\left\{
  \begin{array}{llll}
  1\; ,\; \;
  (g_1,g_2)\in (V_O\times G)\cup (G\times V_O) \\
  x-x_S\; ,\; \;
  (g_1,g_2)\in (U_S\times V_S)\cup (g_0U_T\times V_T) \\
  x-x_T\; ,\; \;
  (g_1,g_2)\in (U_T\times V_T)\cup (g_0U_S\times V_S) \\
  x-x_P\; ,\; \;
  (g_1,g_2)\in V_P\times V_P \\
  \frac{y}{x-x_S}\; ,\; \;
  (g_1,g_2)\in (V_T\times V_P)\cup (U_P\times V_T)\cup (g_0U_P\times V_S) \\
  \frac{y}{x-x_T}\; ,\; \;
  (g_1,g_2)\in (V_S\times V_P)\cup (U_P\times V_S)\cup (g_0U_P\times V_T) \\
  \frac{y}{x-x_P}\; ,\; \;
  (g_1,g_2)\in ((U_S\cup g_0U_S)\times V_T)\cup ((U_T\cup g_0U_T)\times V_S) \\
  \end{array}
  \right.
  $$

  \begin{definition}
  \label{kometa}
  Denote by $B_U$ the cross product defined by the cocycle $\Phi _{O,\psi }$
  (constructed by the cocycle $\phi _{O,\psi }$ of type $III$). Then $B_U$
  is a central simple algebra with center $k(E)$. We shall call $B_U$
  a dihedral cross product because it can be splited by the normal
  extension $\bar k^{U_O}$ with Galois group $G/U_O\cong D_4$.
  \end{definition}

  \begin{proposition}
  \label{atmosfera}
  The dihedral cross product $B_U$ is an unramified central simple
  algebra over $k(E)$ of exponent $2$ and index at most $4$.
  The class of $B_U$ in $\Br (E)$ maps under the homomorphism $\pi $
  into the cohomology class of the cocycle $\phi _{O,\psi}$.
  \end{proposition}
\begin{pf}
Taking into account the considerations in Section \ref{lebeda} one can
calculate
   $$
   \begin{array}{rcl}
   \inv (\cls (\Phi _{O,\psi }))
   &=&
   \beta (\alpha (\cls (\Phi _{O,\psi })))
   \\
   &=&
   \beta (\cls (\di \circ \Phi _{O,\psi }))
   \\
   &=&
   \beta (\cls (\pp \hat {\phi }_{O,\psi }))
   \\
   &=&
   \beta (\delta (\cls ({\phi }_{O,\psi }))
   \\
   &=&
   0\; .
   \end{array}
   $$
Hence $B_U$ is unramified and $\pi ([B_U])= \cls (\phi _{O,\psi
})$.

Denote by $\rho $ the homomorphism $H^2(G,\Gamma )\ra H^2(G,\bar
k(\bar E)^*)$ induced by the composition $\Gamma \stackrel{\gamma
}{\ra }\Omega \subset \bar k(\bar E)^*$. Then
  $$
  \begin{array}{rcl}
  2\cls (\Phi _{O,\psi })
  &=&
  2\cls(\gamma \circ \pp \hat \phi _{O,\psi })
  \\
  &=&
  2\rho \cls(\pp \hat \phi _{O,\psi })
  \\
  &=&
  2\rho \delta (\cls(\phi _{O,\psi }))
  \\
  &=&
  \rho \delta (\cls(2\phi _{O,\psi }))
  \\
  &=&
  0\; .
  \end{array}
  $$
Hence $B_U$ has exponent $2$.

Since the restriction $\Phi _{O,\psi }\palka _{V_O\times V_O}$ is
zero (see explicit formula for $\Phi _{O,\psi }$ below), the algebra
$B_U$ splits by the field $\bar k^{V_O}$. But the subgroup $V_O$ has 
index $4$ in $G$ by Proposition \ref{upiter}. Consequently, 
the index of $B_U$ is at most $4$.
\end{pf}

\subsection{\it End of the proof}
\label{gorgona}

Let $c\in \dv H^1(G,\bar E)$. According to Proposition
\ref{venera} the class $c$ can be presented by a cocycle $\phi
$ of one of three types $I$, $II$ or $III$. Let $U=\ker (\psi )$, 
where $\psi =\phi \palka _H$. If $\phi $ is a cocycle of type $I$, 
then $\pi ([(a,x-x_P)])=c$, where $a\in k$ such that $\bar k^U=k(\sqrt
{a})$ (see Proposition \ref{mcensk}). So one can put
$b_c=[(a,x-x_P)]\in \dv \pi ^{-1}(c)$ (see Section \ref{plane}).
If $\phi $ is a cocycle of type $II$, then $\pi ([(Z,f)])=c$,
where $Z=\bar k^U$ (see Propositions \ref{sochi} and
\ref{suzdol'}) such that $\bar k^U=k(\sqrt {a})$. So one can take
$b_c=[(a,x-x_P)]$. At last, if $\phi $ is a cocycle of type $III$,
then let $b_c=[B_U]$ (see Proposition \ref{atmosfera}). Now
applying Lemma \ref{sech} we finish the proof of Theorem
\ref{main}.

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Dokl. AN Belarusi, 41 (1997), no.6, 13-16.

\bibitem{van den Bergh}
M. van den Bergh {\em The algebraic index of a division algebra.}
Lect. Notes in Math. 1197 (1985) 190-206.

\bibitem[Fadd51]{Fadd}
D.K.Faddeev. {\em Simple algebras over a function field in one
variable.} Proc. Steklov Inst. 38 (1951) 321 - 344; English
transl. in AMS Transl. 3 (1956) 15 - 38.

\bibitem[Mi81]{Milne}
J.S.Milne. {\em Comparison of the Brauer group with the
Tate-\v Safarevi\v c group.} J. Fac. Science,
Univ. Tokyo, Sec. IA. 28 (1981) 735-743 .

\bibitem[Sch69]{Scharlau}
W.Scharlau. {\em \" {U}ber bie Brauer-Gruppe eines algebraischen
Funktionen-k\"{o}rpers in einer Variabein.} J. f\"{u}r die reine
und angew. Math. Bd. 239/240 (1969), S. 1-6.

\bibitem[Serre]{Serre}
J.-P.Serre. {\em Cohomologie Galoisienne.} Springer-Verlag, 1964.

\bibitem[Hall59]{Hall}
M.Hall. {\em The theory of groups.} N.Y. MacMillan, 1959.

\bibitem[Mac63]{Mac}
S.Maclane. {\em Homology.} Die grundlehren der mathematischen
wissenschaften. Springer-Verlag, 1963.

\bibitem[Pie82]{Pie}
R.S.Pierce. {\em Associative algebras.} Graduate Texts in
Mathematics 88, Springer-Verlag, 1982.

\end{thebibliography}


\end{document}

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%
% XYATIP10: upper arrow tips for Xy-pic at 10 point "technical style".
%
% Original Metafont design  Copyright (C) 1991-1997 Kristoffer H. Rose.
% PostScript adaptation     Copyright (C) 1994-1997 Ross Moore.
% Hinting and ATM compatibility  Copyright (C) 1997 Y&Y, Inc.
%
% This file is part of the Xy-pic macro package.
% Xy-pic Copyright (c) 1991-1997  Kristoffer H. Rose  <krisrose@brics.dk>
%
% The Xy-pic macro package is free software; you can redistribute it 
% and/or modify it under the terms of the GNU General Public License 
% as published by the Free Software Foundation; either version 2  
% of the License, or (at your option) any later version.
%
% The Xy-pic macro package is distributed in the hope that it will 
% be useful, but WITHOUT ANY WARRANTY; without even the implied 
% warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  
% See the GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License 
% along with this macro package; if not, write to the 
% Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.

11 dict begin
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cleartomark

%%EndFont 
%%BeginFont: XYBSQL10
%!PS-AdobeFont-1.1: XYBSQL10 001.104
%%CreationDate: 1997 Jul 20 21:19:18
%%RevisionDate: 1997 Sep 17 22:48:36
%
% XYBSQL10: quarter circle segments for Xy-pic at 10 point
%
% Original Metafont design  Copyright (C) 1991-1997 Kristoffer H. Rose.
% PostScript adaptation     Copyright (C) 1994-1997 Ross Moore.
% Hinting and ATM compatibility  Copyright (C) 1997 Y&Y, Inc.
%
% This file is part of the Xy-pic macro package.
% Xy-pic Copyright (c) 1991-1997  Kristoffer H. Rose  <krisrose@brics.dk>
%
% The Xy-pic macro package is free software; you can redistribute it 
% and/or modify it under the terms of the GNU General Public License 
% as published by the Free Software Foundation; either version 2  
% of the License, or (at your option) any later version.
%
% The Xy-pic macro package is distributed in the hope that it will 
% be useful, but WITHOUT ANY WARRANTY; without even the implied 
% warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  
% See the GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License 
% along with this macro package; if not, write to the 
% Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.

11 dict begin
/FontInfo 9 dict dup begin
/version (001.104) readonly def
/Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly def
/FullName (XYBSQL10) readonly def
/FamilyName (XYBSQL) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
/UnderlinePosition -276 def
/UnderlineThickness 138 def
end readonly def
/FontName /XYBSQL10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 31 /d31 put
dup 95 /d95 put
dup 127 /d127 put
readonly def
/FontBBox{-376 -376 376 376}readonly def
/UniqueXX 5092842 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
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1 0 bop 94 98 a Fw(Con)l(ten)l(ts)94 280 y Fv(1)77 b(In)m(tro)s
(duction)2845 b(2)94 463 y(2)77 b(Preliminary)30 b(results)2567
b(4)219 563 y Fu(2.1)84 b Ft(Short)30 b(exact)f(se)l(quenc)l(es)23
b Fu(.)42 b(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f
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f(.)165 b(4)219 662 y(2.2)84 b Ft(L)l(o)l(c)l(al)30 b(invariants)j
Fu(.)42 b(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f
(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)
h(.)f(.)h(.)f(.)165 b(5)219 762 y(2.3)84 b Ft(Principle)32
b(homo)l(gene)l(ous)e(sp)l(ac)l(es)49 b Fu(.)41 b(.)h(.)g(.)f(.)h(.)f
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g(.)f(.)h(.)f(.)h(.)f(.)165 b(5)219 862 y(2.4)84 b Ft(T)-6
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h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)165 b(7)219
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b(7)219 1244 y Fu(3.1)84 b Ft(Starting)29 b(lemmas)84
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f(.)h(.)f(.)165 b(8)219 1343 y(3.2)84 b Ft(F)-6 b(ormula)30
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f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)165 b(8)219 1443 y(3.3)84
b Ft(Classi\014c)l(ation)31 b(of)g(c)l(o)l(cycles)42
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b(10)219 1542 y(3.4)84 b Ft(Dir)l(e)l(ct)29 b(c)l(alculations)i(of)g(c)
l(o)l(cycles)49 b Fu(.)41 b(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h
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124 b(10)219 1642 y(3.5)84 b Fs(\016)33 b Ft(on)d(c)l(o)l(cycles)75
b Fu(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h
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f(.)h(.)f(.)h(.)f(.)124 b(12)94 1825 y Fv(4)77 b(Quaternion)32
b(and)g(cyclic)g(algebras)2032 b(13)219 1924 y Fu(4.1)84
b Ft(A)n(lgebr)l(as)30 b(c)l(orr)l(esp)l(onding)h(to)f(c)l(o)l(cyles)h
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f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)124 b(13)219
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b(pr)l(o)l(ducts)g(c)l(orr)l(esp)l(onding)h(to)f(c)l(o)l(cycles)h(of)f
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b Ft(Cyclic)31 b(algebr)l(as)h(of)e(de)l(gr)l(e)l(e)g
Fu(4)43 b(.)e(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f
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b(16)94 2406 y Fv(5)77 b(Dihedral)32 b(algebras)2588
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31 b(cr)l(oss)f(pr)l(o)l(ducts)g Fu(.)41 b(.)h(.)f(.)h(.)f(.)h(.)g(.)f
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b(21)1845 5697 y(1)p eop
%%Page: 2 2
2 1 bop 279 447 a Fr(Algebras)54 b(of)f(exp)t(onen)l(t)f(2)i(o)l(v)l
(er)e(an)i(elliptic)i(curv)l(e)1624 687 y Fq(V.Guletski)-11
b(\024)-38 b(\020)1502 883 y(Octob)s(er)32 b(19,)g(1999)94
1240 y Fw(1)135 b(In)l(tro)t(duction)94 1422 y Fu(Let)35
b Fs(k)i Fu(b)r(e)e(a)f(\014eld,)h Fs(char)r Fu(\()p
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b(functions)j(on)g Fs(X)7 b Fu(.)36 b(The)27 b(Brauer)e(group)h(Br\()p
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(to)k(the)g(unrami\014ed)94 1721 y(Brauer)h(group)f(Br)699
1733 y Fo(nr)777 1721 y Fu(\()p Fs(k)s Fu(\()p Fs(X)7
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b(This)27 b(allo)n(ws)f(to)i(iden)n(tify)f(Br\()p Fs(X)7
b Fu(\))28 b(with)94 1821 y(Br)186 1833 y Fo(nr)264 1821
y Fu(\()p Fs(k)s Fu(\()p Fs(X)7 b Fu(\))p Fs(=k)s Fu(\).)58
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b(In)36 b([GMY97)o(])f(the)g(description)94 1920 y(of)d(the)h
(2-torsion)c(part)i(in)g(Br)o(\()p Fs(X)7 b Fu(\))32
b(in)f(terms)g(of)h(quaternion)e(algebras)f(w)n(as)i(made)f(under)i
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b(of)26 b(the)g(2-torsion)d(subgroup)h(of)i Fs(J)8 b
Fu(.)36 b(In)26 b([GY98)o(])g(this)f(result)f(w)n(as)h(generated)g(on)g
(algebras)e(of)i(an)94 2119 y(arbitrary)i(exp)r(onen)n(t)i
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2219 y Fs(n)34 b Fu(in)f(Br)o(\()p Fs(X)7 b Fu(\))34
b(can)f(b)r(e)h(presen)n(ted)f(b)n(y)g(a)g(tensor)g(pro)r(duct)h(of)f
(cyclic)f(algebras)e(dep)r(ending)j(on)h(the)g Fs(n)p
Fu(-torsion)94 2319 y(subgroup)27 b(of)g Fs(J)8 b Fu(.)219
2418 y(Let)28 b Fs(X)j Fu(=)23 b Fs(E)34 b Fu(b)r(e)28
b(an)g(elliptic)d(curv)n(e)i(o)n(v)n(er)f Fs(k)s Fu(.)39
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(description)94 2518 y(of)36 b(2-torsion)d(part)i(in)g(Br\()p
Fs(E)5 b Fu(\))36 b(in)f(the)i(case)d(when)i Fs(E)41
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b Fu(.)50 b(Our)32 b(argumen)n(ts)d(based)j(on)g(an)f(explicit)f
(computation)g(of)i(1-co)r(cycles)d(presen)n(ting)h(elemen)n(ts)94
2717 y(of)k(order)e(2)h(in)f(the)i(group)e(of)i(homogeneous)c(spaces)j
(o)n(v)n(er)e Fs(E)39 b Fu(and)33 b(a)g(construction)e(of)j(a)f
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2916 y(It)21 b(should)e(b)r(e)h(men)n(tioned)e(that)j(in)e([YM96],)j
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b(w)n(as)e(made)f(in)i(the)g(case)f(of)h(a)f(lo)r(cal)f(\014eld)h
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(completeness)94 3116 y(prop)r(ert)n(y)e(of)g(the)g(ground)f(\014eld)h
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(a)h(di\013eren)n(t)g(approac)n(h)e(whic)n(h)h(is)g(mainly)94
3215 y(based)k(on)f(cohomology)c(metho)r(ds)k(and)g(allo)n(ws)d(to)k
(consider)d(an)j(arbitrary)c(ground)j(\014eld)f Fs(k)s
Fu(,)i Fs(char)r Fu(\()p Fs(k)s Fu(\))d Fp(6)p Fu(=)e(2)p
Fs(;)14 b Fu(3.)94 3315 y(On)33 b(the)f(other)g(hand)g(w)n(e)g(do)g
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(haracteristic)25 b(of)k(the)g(ground)f(\014eld)g Fs(k)k
Fu(is)c(not)h(2)g(or)f(3,)h(the)h(elliptic)25 b(curv)n(e)j
Fs(E)34 b Fu(can)29 b(b)r(e)h(de\014ned)94 3614 y(as)d(the)h(pro)5
b(jectiv)n(e)26 b(closure)f(of)j(a)f(plane)f(curv)n(e)h(de\014ned)h(b)n
(y)f(an)h(equation)1659 3797 y Fs(y)1703 3762 y Fn(2)1763
3797 y Fu(=)22 b Fs(F)12 b Fu(\()p Fs(x)p Fu(\))24 b
Fs(;)94 3979 y Fu(where)31 b Fs(F)12 b Fu(\()p Fs(x)p
Fu(\))31 b(is)e(a)i(cubic)e(p)r(olynomial)d(o)n(v)n(er)j
Fs(k)k Fu(with)e(the)f(leading)e(co)r(e\016cien)n(t)i(equal)f(to)h(1)g
(without)g(m)n(ultiple)94 4079 y(ro)r(ots)f(in)g(a)g(separable)d
(closure)1125 4057 y(\026)1123 4079 y Fs(k)32 b Fu(of)d
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Fu(.)46 b(Here)30 b Fs(O)k Fu(is)29 b(the)i(in\014nite)e(p)r(oin)n(t)h
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b(of)h(zero)f(on)h(the)g(curv)n(e)f Fs(E)5 b Fu(.)36
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4572 y Fo(P)1825 4560 y Fu(\)\()p Fs(x)h Fp(\000)f Fs(x)2085
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4572 y Fo(S)2447 4560 y Fu(\))94 4743 y(o)n(v)n(er)276
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Fs(E)36 b Fu(denoted)c(b)r(elo)n(w)d(b)n(y)i Fs(P)12
b Fu(.)48 b(In)32 b(other)f(w)n(ords,)f(only)g(one)h(ro)r(ot)g
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b Fu(\()p Fs(x)p Fu(\))33 b(is)e Fs(k)s Fu(-rational.)45
b(If)32 b(all)d(non-trivial)f(p)r(oin)n(ts)i(in)1805
5054 y Fn(2)1861 5021 y Fu(\026)1842 5042 y Fs(E)37 b
Fu(are)30 b(not)i Fs(k)s Fu(-rational,)d(then)j(w)n(e)f(sa)n(y)g(that)h
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b(an)n(y)g(t)n(w)n(o)g(non-trivial)c(2-torsion)i(p)r(oin)n(ts)1845
5697 y(2)p eop
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3 2 bop 94 98 a Fu(on)229 77 y(\026)210 98 y Fs(E)5 b
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b Fu(.)36 b(Then)28 b(an)n(y)f(unrami\014ed)e(cen)n(tral)g(simple)g
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(duct)1323 297 y Fs(A)19 b Fp(\012)f Fu(\()p Fs(b;)c(x)k
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297 y Fu(\))23 b Fs(:)94 447 y Fu(Here)28 b Fs(A)g Fu(is)e(a)h
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Fs(b;)14 b(x)j Fp(\000)g Fs(x)1858 558 y Fo(P)1914 546
y Fu(\),)27 b(\()p Fs(c;)14 b(x)k Fp(\000)e Fs(x)2294
558 y Fo(T)2347 546 y Fu(\))27 b(are)f(quaternion)f(algebras)f(o)n(v)n
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b Fs(b;)14 b(c)25 b Fp(2)g Fs(k)545 616 y Fm(\003)583
646 y Fu(.)41 b(Note)29 b(that)g(\()p Fs(b;)14 b(x)20
b Fp(\000)f Fs(x)1333 658 y Fo(P)1388 646 y Fu(\))30
b(or)e(\()p Fs(c;)14 b(x)19 b Fp(\000)g Fs(x)1855 658
y Fo(T)1908 646 y Fu(\))29 b(can)g(b)r(e)g(trivial)c(in)k(Br)o(\()p
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Fu(is)e(a)94 745 y(square)g(in)g Fs(k)j Fu(then)e(\()p
Fs(b;)14 b(x)19 b Fp(\000)f Fs(x)1016 757 y Fo(P)1072
745 y Fu(\))28 b(is)e(zero)h(in)g(Br)o(\()p Fs(E)5 b
Fu(\).)219 845 y(If)31 b Fs(E)36 b Fu(is)30 b(fully)f(nonsplit,)h(then)
h(the)g(problem)e(of)i(\014nding)e(of)i(all)e(unrami\014ed)f(cen)n
(tral)h(simple)f(algebras)g(of)94 945 y(exp)r(onen)n(t)f(2)g(o)n(v)n
(er)e Fs(k)s Fu(\()p Fs(E)5 b Fu(\))28 b(can)e(b)r(e)i(reduced)e(to)h
(either)f(a)h(split)e(or)h(semisplit)d(case.)36 b(Indeed,)28
b(fully)d(nonsplitness)94 1044 y(means)32 b(that)i Fs(F)45
b Fu(is)32 b(an)h(irreducible)d(p)r(olynomial)e(o)n(v)n(er)k
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Fs(k)s Fu(\()p Fs(x)3060 1056 y Fo(T)3113 1044 y Fu(\))h(has)g(degree)f
(3)94 1144 y(o)n(v)n(er)26 b Fs(k)s Fu(.)37 b(Let)28
b Fs(E)588 1156 y Fo(L)661 1144 y Fu(=)23 b Fs(E)g Fp(\002)18
b Fu(Sp)r(ec\()p Fs(L)p Fu(\).)37 b(The)28 b(comp)r(osition)1331
1336 y(Br)o(\()p Fs(E)5 b Fu(\))1602 1289 y Fn(res)1576
1336 y Fp(\000)-14 b(!)23 b Fu(Br)o(\()p Fs(E)1917 1348
y Fo(L)1967 1336 y Fu(\))2045 1289 y Fn(cor)2023 1336
y Fp(\000)-15 b(!)23 b Fu(Br\()p Fs(E)5 b Fu(\))94 1519
y(coincides)32 b(with)h(the)h(m)n(ultiplication-)o(b)n(y-3)27
b(homomorphism)h(on)34 b(the)g(group)e(Br\()p Fs(E)5
b Fu(\).)55 b(So)34 b(k)n(eeping)d(in)i(mind)94 1619
y(that)28 b(2)g(and)f(3)g(are)g(relativ)n(ely)c(prime)j(w)n(e)h(get)g
(that)h(the)g(corestriction)c(map)1391 1801 y(cor)e(:)1585
1813 y Fn(2)1622 1801 y Fu(Br)o(\()p Fs(E)1806 1813 y
Fo(L)1856 1801 y Fu(\))i Fp(\000)-14 b(!)2082 1813 y
Fn(2)2120 1801 y Fu(Br)o(\()p Fs(E)5 b Fu(\))94 1984
y(is)27 b(surjectiv)n(e.)35 b(Hence)28 b(one)f(can)g(\014nd)h(algebras)
d(in)1734 1996 y Fn(2)1771 1984 y Fu(Br)o(\()p Fs(E)5
b Fu(\))29 b(applying)c(cor)h(to)i(algebras)c(from)3133
1996 y Fn(2)3171 1984 y Fu(Br)o(\()p Fs(E)3355 1996 y
Fo(L)3405 1984 y Fu(\).)219 2084 y(Th)n(us)k(w)n(e)h(see)f(that)h(the)g
(main)d(situation)g(is)i(when)1886 2096 y Fn(2)1923 2084
y Fu(Br)o(\()p Fs(E)5 b Fu(\))29 b(has)f(only)g(one)g(non-trivial)c(p)r
(oin)n(t)k Fs(P)41 b Fu(de\014ned)94 2183 y(o)n(v)n(er)26
b Fs(k)s Fu(.)37 b(In)28 b(this)f(case)1483 2283 y Fs(F)12
b Fu(\()p Fs(x)p Fu(\))24 b(=)f(\()p Fs(x)c Fp(\000)f
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y(\026)1841 4643 y Fs(X)6 b Fu(\))24 b Fp(\000)-14 b(!)23
b Fu(Pic)n(\()2301 4622 y(\026)2277 4643 y Fs(X)7 b Fu(\))23
b Fp(!)g Fu(0)g Fs(;)930 b Fu(\(3\))1190 4812 y(0)22
b Fp(!)i Fu(Pic)1477 4775 y Fn(0)1514 4812 y Fu(\()1570
4791 y(\026)1546 4812 y Fs(X)7 b Fu(\))23 b Fp(\000)-14
b(!)23 b Fu(Pic)n(\()2007 4791 y(\026)1982 4812 y Fs(X)7
b Fu(\))2130 4761 y Fn(deg)2113 4812 y Fp(\000)-14 b(!)23
b Fl(Z)17 b Fp(!)23 b Fu(0)f Fs(:)990 b Fu(\(4\))94 4952
y(Here)29 b(deg)g(is)f(the)h(degree)f(homomorphism)23
b(and)29 b(div)g(is)e(the)j(homomorphism)23 b(sending)28
b Fs(f)33 b Fp(2)3102 4930 y Fu(\026)3100 4952 y Fs(k)s
Fu(\()3202 4931 y(\026)3178 4952 y Fs(X)7 b Fu(\))3286
4921 y Fm(\003)3353 4952 y Fu(in)n(to)27 b(the)94 5051
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5030 y(\026)2433 5051 y Fs(J)f Fu(b)r(e)28 b(the)h(group)e(of)3106
5029 y(\026)3104 5051 y Fs(k)s Fu(-p)r(oin)n(ts)f(of)i(the)94
5151 y(Jacobian)f(v)-5 b(ariet)n(y)28 b Fs(J)38 b Fu(of)29
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(that)f Fs(X)36 b Fu(has)29 b(at)h(least)e(one)h Fs(k)s
Fu(-rational)d(p)r(oin)n(t,)3604 5130 y(\026)3584 5151
y Fs(J)94 5251 y Fu(and)i(Pic)372 5214 y Fn(0)410 5251
y Fu(\()466 5230 y(\026)442 5251 y Fs(X)6 b Fu(\))28
b(are)f(isomorphic)c(as)k(discrete)f Fs(G)p Fu(-mo)r(dules.)35
b(Therefore)26 b(w)n(e)i(ha)n(v)n(e)e(the)i(short)f(exact)g(sequence)
1213 5427 y(0)c Fp(!)1384 5435 y Fu(P)1440 5427 y(\()1496
5406 y(\026)1472 5427 y Fs(X)7 b Fu(\))23 b Fp(\000)-14
b(!)23 b Fu(Div)1891 5391 y Fn(0)1929 5427 y Fu(\()1985
5406 y(\026)1961 5427 y Fs(X)6 b Fu(\))2100 5380 y Fn(sum)2092
5427 y Fp(\000)-15 b(!)2269 5406 y Fu(\026)2248 5427
y Fs(J)32 b Fp(!)23 b Fu(0)f Fs(:)1013 b Fu(\(5\))1845
5697 y(4)p eop
%%Page: 5 5
5 4 bop 94 98 a Fk(2.2)113 b Fj(L)-5 b(o)g(c)g(al)34
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(of)h(co)r(dimension)c(1)j(on)h Fs(X)7 b Fu(\))29 b(b)n(y)h
Fs(X)3375 221 y Fn(\(1\))3463 251 y Fu(.)43 b(All)94
351 y(suc)n(h)32 b(p)r(oin)n(ts)e Fs(x)g Fp(2)f Fs(X)776
321 y Fn(\(1\))896 351 y Fu(are)i(in)f(one-to-one)g(corresp)r(ondence)g
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450 y Fs(k)s Fu(\()p Fs(X)7 b Fu(\))32 b(o)n(v)n(er)e
Fs(k)s Fu(.)49 b(Let)31 b Fs(k)s Fu(\()p Fs(X)7 b Fu(\))950
462 y Fo(x)1023 450 y Fu(b)r(e)32 b(the)g(completion)d(of)i
Fs(k)s Fu(\()p Fs(X)7 b Fu(\))32 b(with)f(resp)r(ect)g(to)h(the)g(v)-5
b(aluation)29 b(corresp)r(onding)94 550 y(to)k(a)f(p)r(oin)n(t)f
Fs(x)h Fp(2)f Fs(X)737 520 y Fn(\(1\))825 550 y Fu(.)52
b(Denote)32 b(b)n(y)g Fs(k)s Fu(\()p Fs(x)p Fu(\))h(the)g(residue)e
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Fs(k)s Fu(\()p Fs(x)p Fu(\))i(with)f(a)g(sub\014eld)g(in)1410
628 y(\026)1408 650 y Fs(k)r Fu(.)43 b(Let)30 b Fs(G)1735
662 y Fo(x)1803 650 y Fu(=)c(Gal)n(\()2057 628 y(\026)2055
650 y Fs(k)t(=k)s Fu(\()p Fs(x)p Fu(\)\))k(for)f(an)n(y)f
Fs(x)f Fp(2)g Fs(X)2882 619 y Fn(\(1\))2970 650 y Fu(.)43
b(Then)29 b(Br\()p Fs(k)s Fu(\()p Fs(X)7 b Fu(\))3564
662 y Fo(x)3606 650 y Fu(\))94 749 y(is)26 b(isomorphic)c(to)k(the)h
(direct)f(sum)f(of)i(the)g(Brauer)e(group)g(Br)o(\()p
Fs(k)s Fu(\()p Fs(x)p Fu(\)\))j(and)f(the)f(group)g(Hom)3042
761 y Fo(cont)3174 749 y Fu(\()p Fs(G)3271 761 y Fo(x)3314
749 y Fs(;)14 b Fl(Q)5 b Fs(=)p Fl(Z)p Fu(\))26 b(of)94
849 y(con)n(tin)n(uous)g(homomorphisms)c(of)28 b(the)g(pro\014nite)e
(group)h Fs(G)2000 861 y Fo(x)2069 849 y Fu(in)n(to)g(the)h(discrete)e
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y(in)n(v)1268 1035 y Fo(x)1333 1023 y Fu(:)23 b(Br)o(\()p
Fs(k)s Fu(\()p Fs(X)7 b Fu(\)\))24 b Fp(\000)-15 b(!)23
b Fu(Hom)2073 1035 y Fo(cont)2206 1023 y Fu(\()p Fs(G)2303
1035 y Fo(x)2346 1023 y Fs(;)14 b Fl(Q)5 b Fs(=)p Fl(Z)p
Fu(\))94 1197 y(b)r(e)28 b(the)g(comp)r(osition)c(of)k(the)g(scalar)d
(extension)h(homomorphism)1414 1371 y(Br)o(\()p Fs(k)s
Fu(\()p Fs(X)7 b Fu(\)\))23 b Fp(\000)-14 b(!)23 b Fu(Br\()p
Fs(k)s Fu(\()p Fs(X)7 b Fu(\))2245 1383 y Fo(x)2286 1371
y Fu(\))94 1545 y(and)28 b(the)g(pro)5 b(jection)648
1719 y(Br\()p Fs(k)s Fu(\()p Fs(X)i Fu(\))958 1731 y
Fo(x)999 1719 y Fu(\))1055 1697 y Fp(\030)1055 1723 y
Fu(=)1142 1719 y(Br\()p Fs(k)s Fu(\()p Fs(x)p Fu(\)\))19
b Fp(\010)g Fu(Hom)1730 1731 y Fo(cont)1863 1719 y Fu(\()p
Fs(G)1960 1731 y Fo(x)2003 1719 y Fs(;)14 b Fl(Q)5 b
Fs(=)p Fl(Z)p Fu(\))22 b Fp(!)h Fu(Hom)2535 1731 y Fo(cont)2668
1719 y Fu(\()p Fs(G)2765 1731 y Fo(x)2807 1719 y Fs(;)14
b Fl(Q)6 b Fs(=)p Fl(Z)p Fu(\))22 b Fs(:)94 1893 y Fu(The)33
b(homomorphism)27 b(in)n(v)973 1905 y Fo(x)1047 1893
y Fu(maps)32 b(a)g(class)e(of)i(algebras)e Fs(a)h Fp(2)g
Fu(Br\()p Fs(k)s Fu(\()p Fs(X)7 b Fu(\)\))32 b(in)n(to)g(its)f
Ft(lo)l(c)l(al)k(invariant)f Fu(in)n(v)3464 1905 y Fo(x)3506
1893 y Fu(\()p Fs(a)p Fu(\).)94 1992 y(The)i(order)d(of)i(in)n(v)710
2004 y Fo(x)752 1992 y Fu(\()p Fs(a)p Fu(\))g(coincides)e(with)h(the)i
(rami\014cation)30 b(index)35 b(of)g(algebras)c(from)j
Fs(a)h Fu(at)g(the)g(p)r(oin)n(t)g Fs(x)g Fp(2)94 2092
y Fs(X)170 2062 y Fn(\(1\))259 2092 y Fu(.)54 b(Algebras)31
b(can)i(b)r(e)h(rami\014ed)d(only)h(in)h(a)g(\014nite)f(n)n(um)n(b)r
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2192 y(homomorphism)1038 2303 y(in)n(v)22 b(:)h(Br)o(\()p
Fs(k)s Fu(\()p Fs(X)7 b Fu(\)\))24 b Fp(\000)-14 b(!)1786
2224 y Fi(M)1740 2411 y Fo(x)p Fm(2)p Fo(X)1882 2394
y Fh(\(1\))1972 2303 y Fu(Hom)2145 2315 y Fo(cont)2278
2303 y Fu(\()p Fs(G)2375 2315 y Fo(x)2417 2303 y Fs(;)14
b Fl(Q)6 b Fs(=)p Fl(Z)p Fu(\))22 b Fs(:)94 2538 y Fu(By)31
b(de\014nition,)e(its)h(k)n(ernel)e(is)i Ft(unr)l(ami\014e)l(d)i(Br)l
(auer)g(gr)l(oup)f Fu(Br)2091 2550 y Fo(nr)2168 2538
y Fu(\()p Fs(k)s Fu(\()p Fs(X)7 b Fu(\)\))31 b(of)g Fs(k)s
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(tioned)94 2638 y(in)e(the)h(b)r(eginning,)e(the)i(group)f(Br)1213
2650 y Fo(nr)1291 2638 y Fu(\()p Fs(k)s Fu(\()p Fs(X)7
b Fu(\)\))29 b(can)f(b)r(e)h(naturally)d(iden)n(ti\014ed)h(with)h(the)h
(Brauer)e(group)h(Br)o(\()p Fs(X)7 b Fu(\))94 2738 y(of)28
b(the)g(curv)n(e)f Fs(X)7 b Fu(.)219 2837 y(Recall)33
b(the)i(cohomology)c(description)h(of)j(the)h(homomorphism)29
b(in)n(v.)59 b(Belo)n(w)34 b Fs(H)2891 2807 y Fm(\003)2928
2837 y Fu(\()p Fs(G;)14 b Fu(?\))36 b(is)e(the)i(Galois)94
2937 y(cohomology)31 b(functor)36 b(on)f(the)g(category)e(of)j
(\(left\))f(discrete)f Fs(G)p Fu(-mo)r(dules.)58 b(By)35
b(Tsen)g(theorem,)g(the)h(group)94 3037 y(Br\()p Fs(k)s
Fu(\()p Fs(X)7 b Fu(\)\))37 b(is)f(isomorphic)c(to)37
b(the)g(cohomology)32 b(group)k Fs(H)2042 3006 y Fn(2)2079
3037 y Fu(\()p Fs(G;)2216 3015 y Fu(\026)2213 3037 y
Fs(k)s Fu(\()2315 3016 y(\026)2291 3037 y Fs(X)7 b Fu(\))2399
3006 y Fm(\003)2437 3037 y Fu(\).)65 b(Let)37 b Fs(M)2796
3049 y Fo(x)2874 3037 y Fu(b)r(e)g(a)g(sum)n(bmo)r(dule)d(in)94
3136 y(Div)q(\()282 3115 y(\026)258 3136 y Fs(X)7 b Fu(\))32
b(generated)e(b)n(y)i(p)r(oin)n(ts)f(in)1279 3115 y(\026)1255
3136 y Fs(X)38 b Fu(mapping)30 b(in)n(to)g Fs(x)h Fp(2)g
Fs(X)2120 3106 y Fn(\(1\))2240 3136 y Fu(under)h(the)g(natural)e(pro)5
b(jection)3345 3115 y(\026)3321 3136 y Fs(X)36 b Fp(!)30
b Fs(X)7 b Fu(.)94 3236 y(The)32 b(discrete)e Fs(G)p
Fu(-mo)r(dule)g Fs(M)1049 3248 y Fo(x)1122 3236 y Fu(is)g(iden)n
(ti\014ed)g(with)h(the)h(induced)f Fs(G)p Fu(-mo)r(dule)e
Fs(M)2707 3199 y Fo(G)2759 3207 y Fg(x)2698 3260 y Fo(G)2800
3236 y Fu(\()p Fl(Z)o Fu(\),)e(where)k Fl(Z)25 b Fu(is)30
b(view)n(ed)94 3345 y(as)c(a)f(trivial)e Fs(G)p Fu(-mo)r(dule.)34
b(Then)26 b(Div)q(\()1328 3324 y(\026)1304 3345 y Fs(X)6
b Fu(\))1435 3323 y Fp(\030)1435 3349 y Fu(=)1522 3283
y Fi(L)1615 3370 y Fo(x)1670 3345 y Fs(M)1760 3308 y
Fo(G)1812 3316 y Fg(x)1751 3369 y Fo(G)1853 3345 y Fu(\()p
Fl(Z)o Fu(\),)21 b(where)k Fs(x)i Fu(runs)e(all)f(p)r(oin)n(ts)h(in)g
Fs(X)3050 3315 y Fn(\(1\))3138 3345 y Fu(.)37 b(Th)n(us)26
b(w)n(e)f(get)375 3540 y Fs(H)451 3506 y Fn(2)488 3540
y Fu(\()p Fs(G;)14 b Fu(Div)q(\()810 3519 y(\026)786
3540 y Fs(X)7 b Fu(\)\))949 3518 y Fp(\030)949 3544 y
Fu(=)1037 3461 y Fi(M)1081 3635 y Fo(x)1176 3540 y Fs(H)1252
3506 y Fn(2)1289 3540 y Fu(\()p Fs(G;)14 b(M)1513 3503
y Fo(G)1565 3511 y Fg(x)1504 3564 y Fo(G)1606 3540 y
Fu(\()p Fl(Z)p Fu(\)\))1781 3518 y Fp(\030)1781 3544
y Fu(=)1869 3461 y Fi(M)1913 3635 y Fo(x)2008 3540 y
Fs(H)2084 3506 y Fn(2)2121 3540 y Fu(\()p Fs(G)2218 3552
y Fo(x)2261 3540 y Fs(;)g Fl(Z)o Fu(\))2408 3518 y Fp(\030)2408
3544 y Fu(=)2496 3461 y Fi(M)2540 3635 y Fo(x)2635 3540
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3552 y Fo(x)3080 3540 y Fs(;)g Fl(Q)6 b Fs(=)p Fl(Z)p
Fu(\))22 b Fs(:)94 3784 y Fu(It)28 b(can)g(b)r(e)g(pro)n(v)n(en)e(that)
i(the)g(comp)r(osition)548 3982 y(Br\()p Fs(k)s Fu(\()p
Fs(X)7 b Fu(\)\))913 3960 y Fp(\030)913 3987 y Fu(=)1001
3982 y Fs(H)1077 3948 y Fn(2)1114 3982 y Fu(\()p Fs(G;)1250
3961 y Fu(\026)1248 3982 y Fs(k)s Fu(\()1350 3961 y(\026)1326
3982 y Fs(X)g Fu(\))1434 3948 y Fm(\003)1472 3982 y Fu(\))1531
3935 y Fn(div)1622 3910 y Ff(\003)1527 3982 y Fp(\000)-14
b(!)23 b Fs(H)1760 3948 y Fn(2)1797 3982 y Fu(\()p Fs(G;)14
b Fu(Div)q(\()2119 3961 y(\026)2095 3982 y Fs(X)7 b Fu(\)\))2258
3960 y Fp(\030)2258 3987 y Fu(=)2346 3904 y Fi(M)2390
4078 y Fo(x)2508 3982 y Fu(Hom)2681 3994 y Fo(cont)2814
3982 y Fu(\()p Fs(G)2911 3994 y Fo(x)2953 3982 y Fs(;)14
b Fl(Q)6 b Fs(=)p Fl(Z)p Fu(\))94 4233 y(coincides)28
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b(induced)30 b(b)n(y)f(div)h(on)g(coho-)94 4333 y(mology)25
b(groups.)35 b(Therefore,)1508 4432 y(Br\()p Fs(X)7 b
Fu(\))1762 4410 y Fp(\030)1762 4436 y Fu(=)1850 4432
y(k)n(er)o(\(div)2107 4395 y Fm(\003)2146 4432 y Fu(\))23
b Fs(:)94 4663 y Fk(2.3)113 b Fj(Principle)34 b(homo)-5
b(gene)g(ous)33 b(sp)-5 b(ac)g(es)94 4816 y Fu(Consider)26
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4956 y Fm(\003)834 4990 y Fu(\))c Fp(!)g Fs(H)1071 4956
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4990 y Fs(k)t Fu(\()1345 4969 y(\026)1321 4990 y Fs(X)6
b Fu(\))1428 4956 y Fm(\003)1467 4990 y Fu(\))1542 4943
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4990 y Fu(\()p Fs(G;)1875 4998 y Fu(P)1932 4990 y(\()1988
4969 y(\026)1964 4990 y Fs(X)7 b Fu(\)\))23 b Fp(!)g
Fs(H)2309 4956 y Fn(3)2346 4990 y Fu(\()p Fs(G;)2483
4969 y Fu(\026)2480 4990 y Fs(k)2526 4956 y Fm(\003)2565
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Fu(\()3076 4969 y(\026)3052 4990 y Fs(X)6 b Fu(\))3159
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(homomorphisms)d(are)k(injectiv)n(e)f(for)h(an)n(y)g
Fs(n)h Fu(\(see)94 5264 y([Sc)n(h69)o(]\).)38 b(Hence)27
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y Fu(5)p eop
%%Page: 6 6
6 5 bop 94 98 a Fu(T)-7 b(aking)24 b(in)n(to)g(accoun)n(t)h(the)h
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361 y Fu(\()p Fs(G;)2304 369 y Fu(P)2361 361 y(\()2417
340 y(\026)2393 361 y Fs(X)6 b Fu(\)\))24 b Fp(!)f Fu(0)f
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Fs(\023)p Fu(\([)p Fs(A)p Fu(]\))25 b(=)e([)p Fs(A)18
b Fp(\012)576 636 y Fo(k)635 624 y Fs(k)s Fu(\()p Fs(X)7
b Fu(\)].)219 753 y(No)n(w)30 b(observ)n(e)f(that)i(div)1007
716 y Fm(\003)1073 753 y Fu(=)c Fs(\014)e Fp(\016)20
b Fs(\013)p Fu(,)32 b(where)e Fs(H)1726 723 y Fn(2)1763
753 y Fu(\()p Fs(G;)1897 761 y Fu(P)1954 753 y(\()2010
732 y(\026)1986 753 y Fs(X)7 b Fu(\)\))2175 702 y Fo(\014)2154
753 y Fp(!)28 b Fs(H)2341 723 y Fn(2)2378 753 y Fu(\()p
Fs(G;)14 b Fu(Div)q(\()2700 732 y(\026)2676 753 y Fs(X)6
b Fu(\)\))32 b(is)d(the)i(homomorphism)94 852 y(induced)22
b(b)n(y)g(the)h(em)n(b)r(edding)1060 860 y(P)1117 852
y(\()1173 831 y(\026)1149 852 y Fs(X)6 b Fu(\))24 b Fp(\032)e
Fu(Div)q(\()1555 831 y(\026)1531 852 y Fs(X)6 b Fu(\).)36
b(Since)21 b Fs(\013)i Fu(is)e(surjectiv)n(e,)h(one)g(has)f(k)n(er)o
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822 y Fm(\000)p Fn(1)3323 852 y Fu(\(k)n(er\()p Fs(\014)t
Fu(\)\).)94 952 y(Consider)j(the)i(exact)f(sequence)905
1140 y Fs(H)981 1105 y Fn(1)1018 1140 y Fu(\()p Fs(G;)14
b Fu(Pic)o(\()1325 1119 y(\026)1301 1140 y Fs(X)6 b Fu(\)\))1503
1093 y Fn(\001)1464 1140 y Fp(\000)-14 b(!)23 b Fs(H)1697
1105 y Fn(2)1734 1140 y Fu(\()p Fs(G;)1868 1148 y Fu(P)1925
1140 y(\()1981 1119 y(\026)1957 1140 y Fs(X)6 b Fu(\)\))2166
1089 y Fo(\014)2120 1140 y Fp(\000)-14 b(!)23 b Fs(H)2353
1105 y Fn(2)2390 1140 y Fu(\()p Fs(G;)14 b Fu(Div)q(\()2712
1119 y(\026)2688 1140 y Fs(X)6 b Fu(\)\))94 1303 y(induced)25
b(b)n(y)f(\(3\).)36 b(It)25 b(follo)n(ws)d(that)j(k)n(er)o(\(div)1469
1266 y Fm(\003)1507 1303 y Fu(\))f(=)e Fs(\013)1703 1273
y Fm(\000)p Fn(1)1793 1303 y Fu(\(im)n(\(\001\)\).)37
b(F)-7 b(urthermore,)24 b(consider)e(the)j(exact)f(sequence)823
1496 y(Pic)o(\()996 1475 y(\026)972 1496 y Fs(X)6 b Fu(\))1079
1461 y Fo(G)1175 1445 y Fn(deg)1158 1496 y Fp(\000)-14
b(!)23 b Fl(Z)17 b Fp(!)23 b Fs(H)1576 1461 y Fn(1)1613
1496 y Fu(\()p Fs(G;)14 b Fu(Pic)1864 1459 y Fn(0)1901
1496 y Fu(\()1957 1475 y(\026)1933 1496 y Fs(X)7 b Fu(\)\))23
b Fp(!)g Fs(H)2278 1461 y Fn(1)2315 1496 y Fu(\()p Fs(G;)14
b Fu(Pic)o(\()2622 1475 y(\026)2598 1496 y Fs(X)7 b Fu(\)\))23
b Fp(!)h Fu(0)94 1659 y(induced)j(b)n(y)f(\(4\).)37 b(Here)26
b(Pic)o(\()1051 1638 y(\026)1027 1659 y Fs(X)6 b Fu(\))1134
1629 y Fo(G)1217 1659 y Fu(is)26 b(the)h(group)f(of)g(divisor)e
(classes)h(in)n(v)-5 b(arian)n(t)23 b(under)k(the)g(action)e(of)i
Fs(G)p Fu(.)37 b(The)94 1759 y(last)28 b(sequence)h(is)e(ended)j(b)n(y)
e(zero)g(b)r(ecause)h Fs(H)1603 1729 y Fn(1)1640 1759
y Fu(\()p Fs(G;)14 b Fl(Z)o Fu(\))20 b(=)25 b(0.)40 b(Since)28
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Fu(,)k(the)g(homomorphism)24 b(deg)30 b(is)94 1858 y(surjectiv)n(e,)j
(hence)f(the)h(epimorphism)28 b Fs(H)1454 1828 y Fn(1)1491
1858 y Fu(\()p Fs(G;)14 b Fu(Pic)1742 1822 y Fn(0)1780
1858 y Fu(\()1836 1837 y(\026)1812 1858 y Fs(X)6 b Fu(\)\))32
b Fp(!)f Fs(H)2173 1828 y Fn(1)2210 1858 y Fu(\()p Fs(G;)14
b Fu(Pic)o(\()2517 1837 y(\026)2493 1858 y Fs(X)7 b Fu(\)\))33
b(is)f(injectiv)n(e.)50 b(Th)n(us)32 b(w)n(e)g(ha)n(v)n(e)94
1958 y(that)1011 2058 y Fs(H)1087 2023 y Fn(1)1124 2058
y Fu(\()p Fs(G;)14 b Fu(Pic)o(\()1431 2037 y(\026)1407
2058 y Fs(X)6 b Fu(\)\))1570 2035 y Fp(\030)1570 2062
y Fu(=)1658 2058 y Fs(H)1734 2023 y Fn(1)1771 2058 y
Fu(\()p Fs(G;)14 b Fu(Pic)2021 2021 y Fn(0)2059 2058
y Fu(\()2115 2037 y(\026)2091 2058 y Fs(X)6 b Fu(\)\))2254
2035 y Fp(\030)2254 2062 y Fu(=)2342 2058 y Fs(H)2418
2023 y Fn(1)2455 2058 y Fu(\()p Fs(G;)2609 2037 y Fu(\026)2589
2058 y Fs(J)i Fu(\))23 b Fs(:)94 2196 y Fu(Then)1422
2295 y(k)n(er)o(\(div)1679 2258 y Fm(\003)1717 2295 y
Fu(\))h(=)e Fs(\013)1913 2261 y Fm(\000)p Fn(1)2003 2295
y Fu(\(im)n(\()p Fs(\016)s Fu(\)\))i Fs(;)94 2433 y Fu(where)31
b Fs(\016)g Fu(:)d Fs(H)533 2403 y Fn(1)570 2433 y Fu(\()p
Fs(G;)724 2412 y Fu(\026)704 2433 y Fs(J)8 b Fu(\))29
b Fp(!)f Fs(H)1006 2403 y Fn(2)1043 2433 y Fu(\()p Fs(G;)1177
2441 y Fu(P)1234 2433 y(\()1290 2412 y(\026)1266 2433
y Fs(X)6 b Fu(\)\))32 b(is)d(the)i(connecting)e(homomorphism)d(induced)
k(b)n(y)g(\(5\).)46 b(Note)31 b(that)94 2533 y(the)g(condition)d
Fs(H)683 2503 y Fn(1)720 2533 y Fu(\()p Fs(G)817 2545
y Fo(x)860 2533 y Fs(;)14 b Fl(Z)o Fu(\))22 b(=)27 b(0)j(implies)c
(that)31 b Fs(H)1719 2503 y Fn(1)1756 2533 y Fu(\()p
Fs(G;)14 b Fu(Div)q(\()2078 2512 y(\026)2054 2533 y Fs(X)6
b Fu(\)\))28 b(=)f(0.)45 b(Then)30 b(the)h(exact)f(sequence)f(induced)
94 2632 y(b)n(y)f(\(3\))g(yields)d(that)j(the)g(connecting)e
(homomorphism)c(\001)28 b(is)f(injectiv)n(e.)35 b(Hence)28
b Fs(\016)i Fu(is)d(injectiv)n(e)f(to)r(o.)36 b(Let)1437
2796 y Fs(\016)1477 2761 y Fm(\000)p Fn(1)1589 2796 y
Fu(:)23 b(im)o(\()p Fs(\016)s Fu(\))g Fp(!)g Fs(H)2037
2761 y Fn(1)2074 2796 y Fu(\()p Fs(G;)2229 2775 y Fu(\026)2208
2796 y Fs(J)9 b Fu(\))94 2959 y(b)r(e)28 b(the)g(in)n(v)n(erse)e(to)h
(the)h(isomorphism)23 b Fs(\016)j Fu(:)d Fs(H)1541 2929
y Fn(1)1578 2959 y Fu(\()p Fs(G;)1732 2938 y Fu(\026)1712
2959 y Fs(J)8 b Fu(\))24 b Fp(!)f Fu(im)n(\()p Fs(\016)s
Fu(\).)37 b(De\014ne)29 b Fs(\031)i Fu(as)c(the)g(comp)r(osition)1003
3160 y Fs(\031)g Fu(:)c(Br)o(\()p Fs(X)7 b Fu(\))1377
3138 y Fp(\030)1377 3164 y Fu(=)1465 3160 y(k)n(er)o(\(div)1722
3123 y Fm(\003)1760 3160 y Fu(\))1861 3113 y Fo(\013)1816
3160 y Fp(\000)-15 b(!)23 b Fu(im)o(\()p Fs(\016)s Fu(\))2204
3113 y Fo(\016)2236 3088 y Ff(\000)p Fh(1)2192 3160 y
Fp(\000)-14 b(!)23 b Fs(H)2425 3126 y Fn(1)2462 3160
y Fu(\()p Fs(G;)2617 3139 y Fu(\026)2596 3160 y Fs(J)9
b Fu(\))23 b Fs(:)803 b Fu(\(6\))94 3323 y(The)28 b(de\014nition)e(of)i
Fs(\031)j Fu(can)c(b)r(e)h(illustrated)c(b)n(y)k(the)g(follo)n(wing)23
b(comm)n(utativ)n(e)h(diagram:)2489 3511 y Fs(H)2565
3480 y Fn(2)2602 3511 y Fu(\()p Fs(G;)14 b Fu(Div)2868
3474 y Fn(0)2905 3511 y Fu(\()2961 3489 y(\026)2937 3510
y Fs(X)3013 3511 y Fu(\)\))2804 3535 y Fe(\177)2824 3556
y(_)2783 3755 y Fd(\017)p Fc(\017)p 2781 3755 4 200 v
655 3848 a Fs(H)731 3818 y Fn(1)768 3848 y Fu(\()p Fs(G;)923
3827 y Fu(\026)902 3848 y Fs(J)9 b Fu(\))993 3804 y Fe(\037)1014
3783 y(\177)1229 3800 y Fo(\016)1476 3825 y Fd(/)p Fc(/)p
1014 3826 463 4 v 1501 3848 a Fs(H)1577 3818 y Fn(2)1614
3848 y Fu(\()p Fs(G;)1748 3856 y Fu(P)1805 3848 y(\()1861
3827 y(\026)1837 3848 y Fs(X)d Fu(\)\))2562 3556 y Fd(4)p
Fc(4)2522 3569 y Fb(i)2484 3581 y(i)2446 3594 y(i)2408
3606 y(i)2370 3619 y(i)2332 3631 y(i)2294 3643 y(i)2256
3656 y(i)2218 3668 y(i)2180 3681 y(i)2142 3693 y(i)2104
3706 y(i)2066 3718 y(i)2028 3730 y(i)1989 3743 y(i)1951
3755 y(i)2222 3890 y Fo(\014)2483 3825 y Fd(/)p Fc(/)p
2002 3826 481 4 v 2507 3848 a Fs(H)2583 3818 y Fn(2)2620
3848 y Fu(\()p Fs(G;)14 b Fu(Div)q(\()2942 3827 y(\026)2918
3848 y Fs(X)7 b Fu(\)\))678 4184 y(k)n(er)o(\(in)n(v\))822
3894 y Fd(O)p Fc(O)p 820 4097 4 203 v 970 4142 a Fe(\037)991
4121 y(\177)1462 4163 y Fd(/)p Fc(/)p 991 4165 472 4
v 1487 4187 a Fs(H)1563 4157 y Fn(2)1600 4187 y Fu(\()p
Fs(G;)1737 4165 y Fu(\026)1734 4187 y Fs(k)s Fu(\()1836
4166 y(\026)1812 4187 y Fs(X)g Fu(\))1920 4157 y Fm(\003)1958
4187 y Fu(\))1764 4006 y Fo(\013)1739 3919 y Fd(O)p Fc(O)1739
3894 y Fd(O)p Fc(O)p 1737 4093 4 200 v 2239 4058 a Fn(in)n(v)2569
3894 y Fd(4)p Fc(4)2529 3907 y Fb(i)2491 3919 y(i)2453
3932 y(i)2414 3944 y(i)2376 3956 y(i)2338 3969 y(i)2299
3981 y(i)2261 3994 y(i)2223 4006 y(i)2184 4019 y(i)2146
4031 y(i)2108 4043 y(i)2069 4056 y(i)2031 4068 y(i)1993
4081 y(i)1954 4093 y(i)706 4519 y Fu(Br\()p Fs(X)g Fu(\))745
4333 y Fm(\030)746 4351 y Fn(=)822 4229 y Fd(O)p Fc(O)p
820 4432 4 203 v 942 4478 a Fe(\037)963 4457 y(\177)1543
4498 y Fd(/)p Fc(/)p 963 4500 581 4 v 1568 4519 a Fu(Br)o(\()p
Fs(k)s Fu(\()p Fs(X)g Fu(\)\))1662 4334 y Fm(\030)1663
4353 y Fn(=)1739 4233 y Fd(O)p Fc(O)p 1737 4432 4 200
v 219 4695 a Fu(No)n(w)27 b(w)n(e)g(ha)n(v)n(e)g(the)h(exact)f
(sequence)1109 4858 y(0)c Fp(!)g Fu(Br)o(\()p Fs(k)s
Fu(\))1559 4811 y Fo(\023)1505 4858 y Fp(\000)-15 b(!)23
b Fu(Br\()p Fs(X)7 b Fu(\))1962 4811 y Fo(\031)1916 4858
y Fp(\000)-15 b(!)24 b Fs(H)2149 4824 y Fn(1)2186 4858
y Fu(\()p Fs(G;)2340 4837 y Fu(\026)2320 4858 y Fs(J)8
b Fu(\))23 b Fp(!)g Fu(0)g Fs(:)909 b Fu(\(7\))94 5021
y(It)33 b(splits.)47 b(Indeed,)33 b(if)e Fs(x)f Fp(2)h
Fs(X)7 b Fu(\()p Fs(k)s Fu(\),)32 b(then)g Fs(k)s Fu(\()p
Fs(x)p Fu(\))f(=)f Fs(k)k Fu(and)e(Br)o(\()p Fs(k)s Fu(\()p
Fs(X)7 b Fu(\))2261 5033 y Fo(x)2303 5021 y Fu(\))2365
4999 y Fp(\030)2365 5026 y Fu(=)2459 5021 y(Br\()p Fs(k)s
Fu(\))21 b Fp(\010)g Fs(H)7 b(om)2957 5033 y Fo(cont)3090
5021 y Fu(\()p Fs(G;)14 b Fl(Q)6 b Fs(=)p Fl(Z)p Fu(\).)48
b(One)94 5121 y(can)28 b(c)n(hec)n(k)e(that)i(the)g(comp)r(osition)567
5284 y(Br)o(\()p Fs(k)s Fu(\))c Fp(!)f Fu(Br)o(\()p Fs(k)s
Fu(\()p Fs(X)7 b Fu(\)\))24 b Fp(!)f Fu(Br)o(\()p Fs(k)s
Fu(\()p Fs(X)7 b Fu(\))1678 5296 y Fo(x)1720 5284 y Fu(\))23
b Fp(!)g Fu(Br\()p Fs(k)s Fu(\))18 b Fp(\010)g Fs(H)7
b(om)2373 5296 y Fo(cont)2506 5284 y Fu(\()p Fs(G;)14
b Fl(Q)6 b Fs(=)p Fl(Z)p Fu(\))22 b Fp(!)h Fu(Br\()p
Fs(k)s Fu(\))94 5448 y(is)k(an)g(iden)n(tical)e(map.)35
b(Hence)28 b Fs(\023)g Fu(has)f(a)h(left)f(in)n(v)n(erse.)1845
5697 y(6)p eop
%%Page: 7 7
7 6 bop 94 98 a Fk(2.4)113 b Fj(T)-7 b(orsion)94 251
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b(!)33 b Fs(B)38 b Fu(is)32 b(a)i(homomorphism)28 b(of)34
b(ab)r(elian)d(groups,)j(then)h(denote)e(b)n(y)94 394
y Fo(n)140 382 y Fs(A)266 342 y Fg(n)307 334 y Fo(\023)232
382 y Fp(\000)-14 b(!)409 394 y Fo(n)455 382 y Fs(B)36
b Fu(and)31 b Fs(A=n)920 327 y Fo(\023=n)903 382 y Fp(\000)-14
b(!)30 b Fs(B)t(=n)h Fu(the)h(homomorphisms)26 b(induced)32
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(the)94 481 y(exact)e(sequence)256 695 y(0)22 b Fp(!)440
707 y Fo(n)486 695 y Fu(Br)o(\()p Fs(k)s Fu(\))745 655
y Fg(n)785 647 y Fo(\023)710 695 y Fp(\000)-14 b(!)881
707 y Fo(n)926 695 y Fu(Br)o(\()p Fs(X)7 b Fu(\))1206
655 y Fg(n)1247 647 y Fo(\031)1180 695 y Fp(\000)-14
b(!)1351 707 y Fo(n)1396 695 y Fs(H)1472 661 y Fn(1)1509
695 y Fu(\()p Fs(G;)1664 674 y Fu(\026)1643 695 y Fs(J)9
b Fu(\))23 b Fp(!)g Fu(Br\()p Fs(k)s Fu(\))p Fs(=n)2192
641 y Fo(\023=n)2175 695 y Fp(\000)-14 b(!)23 b Fu(Br)o(\()p
Fs(X)7 b Fu(\))p Fs(=n)2687 641 y Fo(\031)r(=n)2678 695
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y Fu(\()p Fs(G;)3102 674 y Fu(\026)3082 695 y Fs(J)8
b Fu(\))p Fs(=n)23 b Fp(!)g Fu(0)f Fs(:)94 877 y Fu(If)27
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Fs(\023)p Fu(,)h(then)e(^)-40 b Fs(\023=n)27 b Fu(is)f(a)h(left)g(in)n
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Fs(\023=n)i Fu(is)e(injectiv)n(e.)35 b(Then)27 b(the)h(last)94
977 y(exact)g(sequence)f(yields)e(the)j(short)f(exact)g(sequence)1021
1160 y(0)22 b Fp(!)1205 1172 y Fo(n)1250 1160 y Fu(Br\()p
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1160 y Fp(\000)-14 b(!)1646 1172 y Fo(n)1691 1160 y Fu(Br)o(\()p
Fs(X)7 b Fu(\))1971 1120 y Fg(n)2012 1112 y Fo(\031)1945
1160 y Fp(\000)-14 b(!)2116 1172 y Fo(n)2161 1160 y Fs(H)2237
1125 y Fn(1)2274 1160 y Fu(\()p Fs(G;)2429 1139 y Fu(\026)2408
1160 y Fs(J)9 b Fu(\))23 b Fp(!)g Fu(0)g Fs(:)94 1342
y Fu(W)-7 b(e)28 b(shall)e(use)h(the)h(follo)n(wing)c(simple)94
1508 y Fv(Lemma)30 b(2.4.1)40 b Ft(L)l(et)33 b Fu(0)c
Fp(!)h Fs(A)1149 1461 y Fo(\023)1095 1508 y Fp(\000)-15
b(!)31 b Fs(B)1402 1461 y Fo(\031)1356 1508 y Fp(\000)-15
b(!)30 b Fs(C)37 b Fp(!)30 b Fu(0)j Ft(b)l(e)g(an)h(exact)g(se)l(quenc)
l(e.)49 b(Supp)l(ose)34 b(that)g(the)g(homomor-)94 1608
y(phism)d Fs(\023)f Ft(has)h(a)f(left)g(inverse)f Fu(^)-41
b Fs(\023)24 b Fu(:)f Fs(B)k Fp(!)c Fs(C)6 b Ft(.)39
b(F)-6 b(or)30 b(any)g Fs(c)23 b Fp(2)g Fs(C)36 b Ft(cho)l(ose)31
b Fs(b)2312 1620 y Fo(c)2369 1608 y Fp(2)23 b Fs(\031)2497
1578 y Fm(\000)p Fn(1)2586 1608 y Fu(\()p Fs(c)p Fu(\))p
Ft(.)40 b(Then)30 b(any)g(element)g Fs(b)22 b Fp(2)i
Fs(B)94 1708 y Ft(has)35 b(the)f(unique)g(de)l(c)l(omp)l(osition)i
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b Fs(\023)p Fu(\()p Fs(a)p Fu(\))p Ft(,)36 b(wher)l(e)f
Fs(c)c Fu(=)f Fs(\031)s Fu(\()p Fs(b)p Fu(\))i Fp(2)f
Fs(C)40 b Ft(and)35 b Fs(a)30 b Fu(=)h Fs(\023)2868 1678
y Fm(\000)p Fn(1)2957 1708 y Fu(\()p Fs(b)22 b Fp(\000)f
Fs(b)3169 1720 y Fo(c)3202 1708 y Fu(\))p Ft(.)52 b(The)35
b(map)99 1807 y Fu(^)-47 b Fs(\031)27 b Fu(:)c Fs(C)29
b Fp(!)23 b Fs(B)t Ft(,)31 b Fs(c)23 b Fp(7!)g Fs(b)732
1819 y Fo(c)784 1807 y Fp(\000)18 b Fs(\023)o Fu(^)-41
b Fs(\023)p Fu(\()p Fs(b)993 1819 y Fo(c)1027 1807 y
Fu(\))30 b Ft(is)g(a)g(right)h(inverse)f(homomorphism)j(of)d
Fs(\031)s Ft(.)94 2040 y Fk(2.5)113 b Fj(The)34 b(plane)g(of)h(c)-5
b(omputations)94 2193 y Fu(No)n(w)28 b(let)f Fs(X)i Fu(=)23
b Fs(E)32 b Fu(b)r(e)c(an)g(elliptic)c(curv)n(e)i(o)n(v)n(er)g
Fs(k)s Fu(.)37 b(Then)1926 2172 y(\026)1906 2193 y Fs(J)31
b Fu(=)2090 2172 y(\026)2070 2193 y Fs(E)i Fu(and)28
b(there)f(exists)f(the)i(split)e(exact)h(sequence)1031
2376 y(0)c Fp(!)1216 2388 y Fn(2)1253 2376 y Fu(Br)o(\()p
Fs(k)s Fu(\))1516 2336 y Fh(2)1548 2328 y Fo(\023)1478
2376 y Fp(\000)-15 b(!)1648 2388 y Fn(2)1686 2376 y Fu(Br)o(\()p
Fs(E)5 b Fu(\))1961 2336 y Fh(2)1993 2328 y Fo(\031)1931
2376 y Fp(\000)-15 b(!)2101 2388 y Fn(2)2139 2376 y Fs(H)2215
2341 y Fn(1)2252 2376 y Fu(\()p Fs(G;)2405 2355 y Fu(\026)2386
2376 y Fs(E)5 b Fu(\))23 b Fp(!)g Fu(0)g Fs(:)94 2558
y Fu(F)-7 b(or)33 b(ev)n(ery)e(cohomology)e(class)i Fs(c)g
Fp(2)1301 2570 y Fn(2)1339 2558 y Fs(H)1415 2528 y Fn(1)1452
2558 y Fu(\()p Fs(G;)1605 2537 y Fu(\026)1586 2558 y
Fs(E)5 b Fu(\))33 b(w)n(e)g(w)n(an)n(t)f(to)h(c)n(ho)r(ose)e(a)i
(Brauer)e(class)g Fs(b)3020 2570 y Fo(c)3085 2558 y Fp(2)3186
2570 y Fn(2)3223 2558 y Fu(Br\()p Fs(E)5 b Fu(\))33 b(suc)n(h)94
2658 y(that)293 2670 y Fn(2)331 2658 y Fs(\031)s Fu(\()p
Fs(b)449 2670 y Fo(c)483 2658 y Fu(\))f(=)f Fs(c)p Fu(.)52
b(W)-7 b(e)34 b(shall)c(describ)r(e)h(classes)g Fs(c)h
Fp(2)1866 2670 y Fn(2)1904 2658 y Fs(H)1980 2628 y Fn(1)2017
2658 y Fu(\()p Fs(G;)2170 2637 y Fu(\026)2151 2658 y
Fs(E)5 b Fu(\))33 b(in)f(terms)g(of)h(explicit)d(co)r(cycles)h
(\(Section)94 2758 y(3\).)56 b(After)34 b(that)g(w)n(e)g(construct)f
(cen)n(tral)f(simple)f(algebras)g(represen)n(ting)g(the)j(Brauer)f
(classes)e Fs(b)3245 2770 y Fo(c)3313 2758 y Fu(in)i(Br)o(\()p
Fs(E)5 b Fu(\))94 2857 y(\(Sections)32 b(4)f(and)h(5\).)51
b(Since)31 b(b)n(y)h(Lemma)e(2.4.1)h(an)n(y)h(elemen)n(t)e
Fs(b)g Fp(2)2326 2869 y Fn(2)2363 2857 y Fu(Br\()p Fs(E)5
b Fu(\))33 b(has)e(a)h(unique)f(decomp)r(osition)94 2957
y Fs(b)e Fu(=)266 2969 y Fn(2)303 2957 y Fs(\023)p Fu(\()p
Fs(a)p Fu(\))22 b(+)e Fs(b)583 2969 y Fo(c)616 2957 y
Fu(,)32 b(where)f Fs(a)d Fp(2)1085 2969 y Fn(2)1123 2957
y Fu(Br)o(\()p Fs(k)s Fu(\))j(and)g Fs(c)e Fp(2)1683
2969 y Fn(2)1720 2957 y Fs(H)1796 2927 y Fn(1)1833 2957
y Fu(\()p Fs(G;)1987 2936 y Fu(\026)1967 2957 y Fs(E)5
b Fu(\),)33 b(w)n(e)d(get)h(that)g(a)g(full)f(description)e(of)j
(2-torsion)94 3056 y(part)d(of)f(Br)o(\()p Fs(E)5 b Fu(\))p
Fs(=)p Fu(Br\()p Fs(k)s Fu(\))28 b(is)f(giv)n(en)e(b)n(y)j(cen)n(tral)d
(simple)g(algebras)g(represen)n(ting)g(classes)g Fs(b)2904
3068 y Fo(c)2937 3056 y Fu(.)219 3156 y(The)j(Kummer)d(sequence)1391
3273 y(0)e Fp(!)1575 3285 y Fn(2)1632 3252 y Fu(\026)1613
3273 y Fs(E)28 b Fp(\000)-14 b(!)1878 3252 y Fu(\026)1859
3273 y Fs(E)1998 3226 y Fn(2)1948 3273 y Fp(\000)g(!)2124
3252 y Fu(\026)2105 3273 y Fs(E)28 b Fp(!)23 b Fu(0)1190
b(\(8\))94 3423 y(yields)26 b(the)i(exact)f(sequence)966
3605 y(0)22 b Fp(!)i Fs(E)5 b Fu(\()p Fs(k)s Fu(\))p
Fs(=)p Fu(2)22 b Fp(\000)-14 b(!)23 b Fs(H)1652 3571
y Fn(1)1689 3605 y Fu(\()p Fs(G;)1837 3617 y Fn(2)1894
3584 y Fu(\026)1874 3605 y Fs(E)6 b Fu(\))23 b Fp(\000)-14
b(!)2167 3617 y Fn(2)2204 3605 y Fs(H)2280 3571 y Fn(1)2317
3605 y Fu(\()p Fs(G;)2470 3584 y Fu(\026)2451 3605 y
Fs(E)5 b Fu(\))24 b Fp(!)f Fu(0)f Fs(:)766 b Fu(\(9\))94
3788 y(It)29 b(follo)n(ws)24 b(that)29 b(an)n(y)e(elemen)n(t)f(in)1212
3800 y Fn(2)1250 3788 y Fs(H)1326 3758 y Fn(1)1363 3788
y Fu(\()p Fs(G;)1516 3767 y Fu(\026)1497 3788 y Fs(E)5
b Fu(\))28 b(can)g(b)r(e)g(presen)n(ted)f(b)n(y)h(a)f(co)r(cycle)f
Fs(\036)e Fu(:)g Fs(G)g Fp(!)3061 3800 y Fn(2)3117 3767
y Fu(\026)3098 3788 y Fs(E)33 b Fu(of)28 b(the)g(group)94
3888 y Fs(G)34 b Fu(with)f(co)r(e\016cien)n(ts)f(in)931
3900 y Fn(2)987 3867 y Fu(\026)968 3888 y Fs(E)5 b Fu(.)55
b(Moreo)n(v)n(er,)32 b(the)i(cohomology)29 b(class)j(cls)o(\()p
Fs(\036)p Fu(\))h Fp(2)2659 3900 y Fn(2)2696 3888 y Fs(H)2772
3858 y Fn(1)2809 3888 y Fu(\()p Fs(G;)2962 3867 y Fu(\026)2943
3888 y Fs(E)5 b Fu(\))34 b(is)f(trivial)c(i\013)k(the)94
3987 y(cohomology)28 b(class)h(cls)o(\()p Fs(\036)p Fu(\))h
Fp(2)g Fs(H)1148 3957 y Fn(1)1185 3987 y Fu(\()p Fs(G;)1333
3999 y Fn(2)1390 3966 y Fu(\026)1370 3987 y Fs(E)6 b
Fu(\))31 b(lies)f(in)g(the)i(image)d(of)i(the)h(em)n(b)r(edding)d
Fs(E)5 b Fu(\()p Fs(k)s Fu(\))p Fs(=)p Fu(2)29 b Fs(,)-14
b Fp(!)30 b Fs(H)3294 3957 y Fn(1)3331 3987 y Fu(\()p
Fs(G;)3479 3999 y Fn(2)3536 3966 y Fu(\026)3516 3987
y Fs(E)5 b Fu(\).)94 4087 y(This)25 b(suggests)e(that)j(\014rstly)d(w)n
(e)i(need)g(to)g(describ)r(e)f(all)f(1-co)r(cycles)f(of)j(the)h(group)e
Fs(G)h Fu(with)g(co)r(e\016cien)n(ts)e(in)3511 4099 y
Fn(2)3568 4066 y Fu(\026)3549 4087 y Fs(E)5 b Fu(.)94
4187 y(It)34 b(will)c(pro)n(vide)g(the)j(description)e(of)1344
4199 y Fn(2)1381 4187 y Fs(H)1457 4156 y Fn(1)1494 4187
y Fu(\()p Fs(G;)1648 4166 y Fu(\026)1628 4187 y Fs(E)6
b Fu(\))33 b(in)f(terms)g(of)g(explicit)f(1-co)r(cycles)f(of)i(the)i
(group)d Fs(G)i Fu(with)94 4286 y(co)r(e\016cien)n(ts)27
b(in)625 4298 y Fn(2)682 4265 y Fu(\026)663 4286 y Fs(E)5
b Fu(.)94 4561 y Fw(3)135 b(Calculus)44 b(of)h(co)t(cycles)94
4743 y Fu(Let)38 b Fs(E)43 b Fu(b)r(e)38 b(a)f(semisplit)d(elliptic)g
(curv)n(e)j(and)h(let)e Fp(f)p Fs(O)r(;)14 b(P)e Fp(g)38
b Fu(b)r(e)g(the)g(group)e Fs(k)s Fu(-rational)e(2-torsion)h(p)r(oin)n
(ts)h(on)94 4842 y Fs(E)5 b Fu(.)67 b(Then)38 b Fs(F)12
b Fu(\()p Fs(x)p Fu(\))40 b(=)f(\()p Fs(x)26 b Fp(\000)e
Fs(x)1038 4854 y Fo(P)1094 4842 y Fu(\))p Fs(f)9 b Fu(\()p
Fs(x)p Fu(\),)41 b(where)c Fs(f)9 b Fu(\()p Fs(x)p Fu(\))38
b(is)e(an)i(irreducible)33 b(quadratic)j(p)r(olynomial)d(o)n(v)n(er)i
Fs(k)s Fu(.)66 b(If)94 4954 y Fn(2)132 4942 y Fs(E)31
b Fu(=)26 b Fp(f)p Fs(O)r(;)14 b(P)r(;)g(T)7 b(;)14 b(S)5
b Fp(g)28 b Fu(is)g(the)i(group)e(of)i(all)d(2-torsion)f(p)r(oin)n(ts)j
(on)2187 4921 y(\026)2168 4942 y Fs(E)34 b Fu(then)c
Fs(f)9 b Fu(\()p Fs(x)p Fu(\))27 b(=)f(\()p Fs(x)20 b
Fp(\000)f Fs(x)2963 4954 y Fo(T)3016 4942 y Fu(\)\()p
Fs(x)i Fp(\000)e Fs(x)3279 4954 y Fo(S)3328 4942 y Fu(\))29
b(o)n(v)n(er)3571 4920 y(\026)3569 4942 y Fs(k)s Fu(.)94
5041 y(Let)h Fs(L=k)h Fu(b)r(e)f(the)f(minimal)c(sub)r(extension)j(in)
1598 5020 y(\026)1596 5041 y Fs(k)s(=k)k Fu(suc)n(h)c(that)i(all)d
(2-torsion)f(p)r(oin)n(ts)i(of)h Fs(E)35 b Fu(are)28
b(rational)e(o)n(v)n(er)94 5141 y Fs(L)p Fu(,)i(i.e.)36
b Fs(L)22 b Fu(=)h Fs(k)s Fu(\()p Fs(x)637 5153 y Fo(T)690
5141 y Fu(\))g(=)g Fs(k)s Fu(\()p Fs(x)958 5153 y Fo(S)1006
5141 y Fu(\).)38 b(The)27 b(group)g Fs(H)j Fu(=)22 b(Gal)o(\()1855
5119 y(\026)1853 5141 y Fs(k)s(=L)p Fu(\))27 b(has)g(index)g(2)g(in)g
Fs(G)p Fu(.)1845 5697 y(7)p eop
%%Page: 8 8
8 7 bop 94 98 a Fk(3.1)113 b Fj(Starting)35 b(lemmas)94
251 y Fu(Consider)26 b(the)i(exact)f(sequence)527 454
y(0)c Fp(!)g Fs(H)774 420 y Fn(1)811 454 y Fu(\()p Fs(G=H)r(;)1072
466 y Fn(2)1128 433 y Fu(\026)1109 454 y Fs(E)5 b Fu(\))1257
407 y Fn(inf)1231 454 y Fp(\000)-15 b(!)23 b Fs(H)1463
420 y Fn(1)1500 454 y Fu(\()p Fs(G;)1649 466 y Fn(2)1705
433 y Fu(\026)1686 454 y Fs(E)5 b Fu(\))1834 407 y Fn(inf)1807
454 y Fp(\000)-14 b(!)23 b Fs(H)2040 420 y Fn(1)2077
454 y Fu(\()p Fs(H)r(;)2231 466 y Fn(2)2288 433 y Fu(\026)2269
454 y Fs(E)5 b Fu(\))2367 420 y Fo(G=H)2539 454 y Fp(\000)-15
b(!)23 b Fs(H)2771 420 y Fn(2)2808 454 y Fu(\()p Fs(G=H)r(;)3069
466 y Fn(2)3126 433 y Fu(\026)3107 454 y Fs(E)5 b Fu(\))94
637 y(induced)31 b(b)n(y)g(the)g(Ho)r(c)n(hsc)n(hild-Serre)d(sp)r
(ectral)h(sequence)i Fs(H)2048 607 y Fo(p)2086 637 y
Fu(\()p Fs(G=H)r(;)14 b(H)2409 607 y Fo(q)2445 637 y
Fu(\()p Fs(H)r(;)2600 649 y Fn(2)2656 616 y Fu(\026)2637
637 y Fs(E)5 b Fu(\)\))29 b Fp(\))g Fs(H)2984 607 y Fo(n)3029
637 y Fu(\()p Fs(G;)3178 649 y Fn(2)3234 616 y Fu(\026)3215
637 y Fs(E)5 b Fu(\))31 b(\(see,)h(for)94 737 y(example,)d([Serre)o
(]\).)44 b(It)30 b(is)f(easy)g(to)g(sho)n(w)g(that)i
Fs(H)1709 706 y Fn(1)1746 737 y Fu(\()p Fs(G=H)r(;)2007
749 y Fn(2)2063 716 y Fu(\026)2044 737 y Fs(E)5 b Fu(\))27
b(=)g(0)i(and)h Fs(H)2572 706 y Fn(2)2609 737 y Fu(\()p
Fs(G=H)r(;)2870 749 y Fn(2)2926 716 y Fu(\026)2907 737
y Fs(E)5 b Fu(\))27 b(=)g(0.)43 b(So)30 b(w)n(e)f(ha)n(v)n(e)94
836 y(the)f(isomorphism)1193 952 y(res)23 b(:)g Fs(H)1441
918 y Fn(1)1478 952 y Fu(\()p Fs(G;)1626 964 y Fn(2)1682
931 y Fu(\026)1663 952 y Fs(E)5 b Fu(\))1826 891 y Fm(\030)1826
909 y Fn(=)1785 952 y Fp(\000)-15 b(!)23 b Fs(H)2017
918 y Fn(1)2054 952 y Fu(\()p Fs(H)r(;)2209 964 y Fn(2)2265
931 y Fu(\026)2246 952 y Fs(E)5 b Fu(\))2344 918 y Fo(G=H)2516
952 y Fs(:)951 b Fu(\(10\))94 1102 y(Recall)32 b(the)j(action)e(of)h
Fs(G=H)41 b Fu(on)34 b Fs(H)1273 1072 y Fn(1)1310 1102
y Fu(\()p Fs(H)r(;)1464 1114 y Fn(2)1520 1081 y Fu(\026)1501
1102 y Fs(E)5 b Fu(\).)57 b(Let)35 b Fs( )i Fp(2)d Fs(H)2091
1072 y Fn(1)2128 1102 y Fu(\()p Fs(H)r(;)2282 1114 y
Fn(2)2339 1081 y Fu(\026)2320 1102 y Fs(E)5 b Fu(\))34
b(=)g Fs(H)7 b(om)p Fu(\()p Fs(H)r(;)2894 1114 y Fn(2)2950
1081 y Fu(\026)2931 1102 y Fs(E)e Fu(\))35 b(and)f Fs(g)3272
1114 y Fn(0)3343 1102 y Fp(2)g Fs(G)p Fp(n)p Fs(H)7 b
Fu(.)94 1201 y(Then)36 b(\()p Fs(g)391 1213 y Fn(0)428
1201 y Fs( )s Fu(\)\()p Fs(g)s Fu(\))h(=)f Fs(g)802 1213
y Fn(0)838 1201 y Fu(\()p Fs( )s Fu(\()p Fs(g)1002 1166
y Fm(\000)p Fn(1)999 1224 y(0)1092 1201 y Fs(g)s(g)1175
1213 y Fn(0)1211 1201 y Fu(\)\))g(for)f(an)n(y)g Fs(g)k
Fp(2)d Fs(G)p Fu(.)61 b(Note)35 b(that)h(this)f(action)f(do)r(es)h(not)
g(dep)r(end)h(on)f(the)94 1301 y(c)n(hoice)26 b(of)h
Fs(g)475 1313 y Fn(0)539 1301 y Fu(and)g(w)n(e)f(can)h(consider)e(the)i
(action)f(of)h Fs(g)1821 1313 y Fn(0)1885 1301 y Fu(as)f(an)h(action)e
(of)i(the)h(unique)e(non)h(trivial)c(elemen)n(t)j(in)94
1401 y Fs(G=H)7 b Fu(.)219 1500 y(No)n(w)27 b(\014x)h(an)f(isomorphism)
1449 1612 y Fn(2)1505 1579 y Fu(\026)1486 1600 y Fs(E)1575
1578 y Fp(\030)1575 1604 y Fu(=)1663 1600 y Fp(f)p Fs(O)r(;)14
b(T)e Fp(g)17 b(\010)h(f)p Fs(O)r(;)c(S)5 b Fp(g)22 b
Fs(:)94 1749 y Fu(Then)893 1849 y Fs(H)7 b(om)p Fu(\()p
Fs(H)r(;)1236 1861 y Fn(2)1292 1828 y Fu(\026)1273 1849
y Fs(E)e Fu(\))1394 1827 y Fp(\030)1394 1853 y Fu(=)1482
1849 y Fs(H)i(om)p Fu(\()p Fs(H)r(;)14 b Fp(f)p Fs(O)r(;)g(T)e
Fp(g)p Fu(\))18 b Fp(\010)g Fs(H)7 b(om)p Fu(\()p Fs(H)r(;)14
b Fp(f)p Fs(O)r(;)g(S)5 b Fp(g)p Fu(\))22 b Fs(:)94 1998
y Fu(Let)i(\()p Fs( )325 2010 y Fn(1)362 1998 y Fs(;)14
b( )453 2010 y Fn(2)490 1998 y Fu(\))24 b(b)r(e)f(an)g(elemen)n(t)e(in)
h Fs(H)7 b(om)p Fu(\()p Fs(H)r(;)14 b Fp(f)p Fs(O)r(;)g(T)e
Fp(g)p Fu(\))d Fp(\010)g Fs(H)e(om)p Fu(\()p Fs(H)r(;)14
b Fp(f)p Fs(O)r(;)g(S)5 b Fp(g)p Fu(\))22 b(and)h Fs(U)2689
2010 y Fo(i)2739 1998 y Fu(=)g(k)n(er)o(\()p Fs( )3024
2010 y Fo(i)3052 1998 y Fu(\),)h Fs(i)f Fu(=)f(1)p Fs(;)14
b Fu(2.)35 b(Then)94 2098 y(the)28 b(pair)e(\()p Fs( )494
2110 y Fn(1)531 2098 y Fs(;)14 b( )622 2110 y Fn(2)660
2098 y Fu(\))27 b(lies)e(in)i(\()p Fs(H)7 b(om)p Fu(\()p
Fs(H)r(;)14 b Fp(f)p Fs(O)r(;)g(T)e Fp(g)p Fu(\))17 b
Fp(\010)g Fs(H)7 b(om)p Fu(\()p Fs(H)r(;)14 b Fp(f)p
Fs(O)r(;)g(S)5 b Fp(g)p Fu(\)\))2333 2068 y Fo(G=H)2509
2098 y Fu(if)27 b(and)g(only)f(if)h Fs(U)3061 2110 y
Fn(2)3121 2098 y Fu(=)c Fs(g)3252 2063 y Fm(\000)p Fn(1)3249
2120 y(0)3340 2098 y Fs(U)3397 2110 y Fn(1)3434 2098
y Fs(g)3474 2110 y Fn(0)3538 2098 y Fu(for)94 2198 y(some)e
Fs(g)337 2210 y Fn(0)397 2198 y Fp(2)j Fs(G)p Fp(n)p
Fs(H)k Fu(\(this)22 b(condition)e(do)r(es)i(not)h(dep)r(end)g(on)f(the)
g(c)n(hoice)f(of)h(an)g(elemen)n(t)e Fs(g)2865 2210 y
Fn(0)2925 2198 y Fu(in)h Fs(G)p Fp(n)p Fs(H)7 b Fu(\).)35
b(Therefore)94 2297 y(one)24 b(can)f(iden)n(tify)g(the)h(set)g(of)f
(non)n(trivial)d(homomorphisms)e(in)23 b(\()p Fs(H)7
b(om)p Fu(\()p Fs(H)r(;)14 b Fp(f)p Fs(O)r(;)g(T)e Fp(g)p
Fu(\))f Fp(\010)g Fs(H)c(om)p Fu(\()p Fs(H)r(;)14 b Fp(f)p
Fs(O)r(;)g(S)5 b Fp(g)p Fu(\)\))3490 2267 y Fo(G=H)94
2397 y Fu(with)36 b(the)h(set)f(of)g(subgroups)f Fs(U)45
b Fu(of)36 b(index)f(t)n(w)n(o)h(in)f Fs(H)7 b Fu(.)63
b(A)36 b(suc)n(h)g Fs(U)45 b Fu(corresp)r(onds)34 b(to)i(the)h
(homomorphism)94 2497 y Fs( )28 b Fu(=)c Fs( )319 2509
y Fo(T)391 2497 y Fu(+)19 b Fs( )529 2509 y Fo(S)577
2497 y Fu(,)29 b(where)f Fs( )924 2509 y Fo(T)1005 2497
y Fu(is)f(the)i(surjectiv)n(e)e(homomorphism)d Fs(H)31
b Fp(!)25 b(f)p Fs(O)r(;)14 b(T)e Fp(g)27 b Fu(with)h(the)h(k)n(ernel)e
Fs(U)9 b Fu(,)28 b(and)h Fs( )3590 2509 y Fo(S)94 2596
y Fu(is)e(the)h(surjectiv)n(e)e(homomorphism)c Fs(H)30
b Fp(!)23 b(f)p Fs(O)r(;)14 b(S)5 b Fp(g)27 b Fu(with)g(the)h(k)n
(ernel)e Fs(g)2373 2561 y Fm(\000)p Fn(1)2370 2618 y(0)2462
2596 y Fs(U)9 b(g)2568 2608 y Fn(0)2604 2596 y Fu(.)94
2779 y Fv(Lemma)30 b(3.1.1)40 b Ft(If)30 b Fs(\036)23
b Fu(:)h Fs(G)f Fp(!)1085 2791 y Fn(2)1141 2758 y Fu(\026)1122
2779 y Fs(E)35 b Ft(is)30 b(a)g(c)l(o)l(cycle,)i(then)d
Fs(V)42 b Fu(=)23 b(k)n(er)o(\()p Fs(\036)-5 b Fp(j)2255
2791 y Fo(H)2319 2779 y Fu(\))30 b Ft(is)g(a)g(normal)g(sub)l(gr)l(oup)
g(in)g Fs(G)p Ft(.)94 2961 y(Pr)l(o)l(of)p Fu(.)73 b(Isomorphism)35
b(\(10\))j(sho)n(ws)g(that)i Fs(\036)14 b Fp(j)1600 2973
y Fo(H)1703 2961 y Fu(is)37 b(in)n(v)-5 b(arian)n(t)36
b(under)j(the)h(action)d(of)i Fs(G=H)7 b Fu(.)71 b(Then)39
b(either)94 3061 y Fs(V)46 b Fu(=)26 b Fs(H)37 b Fu(or)29
b Fs(V)45 b Fu(=)27 b Fs(U)h Fp(\\)21 b Fs(g)878 3026
y Fm(\000)p Fn(1)875 3083 y(0)966 3061 y Fs(U)9 b(g)1072
3073 y Fn(0)1139 3061 y Fu(for)29 b(some)f(subgroup)h
Fs(U)39 b Fu(of)29 b(index)g(2)h(in)f Fs(H)7 b Fu(.)43
b(In)30 b(the)g(second)f(case)g Fs(g)3314 3026 y Fm(\000)p
Fn(1)3311 3083 y(0)3403 3061 y Fs(V)19 b(g)3510 3073
y Fn(0)3573 3061 y Fu(=)94 3161 y Fs(g)137 3125 y Fm(\000)p
Fn(1)134 3183 y(0)226 3161 y Fu(\()p Fs(U)24 b Fp(\\)15
b Fs(g)452 3125 y Fm(\000)p Fn(1)449 3183 y(0)540 3161
y Fs(U)9 b(g)646 3173 y Fn(0)683 3161 y Fu(\))p Fs(g)755
3173 y Fn(0)815 3161 y Fu(=)23 b Fs(g)946 3125 y Fm(\000)p
Fn(1)943 3183 y(0)1034 3161 y Fs(U)9 b(g)1140 3173 y
Fn(0)1192 3161 y Fp(\\)15 b Fs(g)1305 3125 y Fm(\000)p
Fn(2)1302 3183 y(0)1393 3161 y Fs(U)9 b(g)1502 3131 y
Fn(2)1499 3181 y(0)1562 3161 y Fu(=)22 b Fs(g)1692 3125
y Fm(\000)p Fn(1)1689 3183 y(0)1781 3161 y Fs(U)9 b(g)1887
3173 y Fn(0)1938 3161 y Fp(\\)15 b Fs(U)32 b Fu(=)23
b Fs(V)c Fu(.)36 b(F)-7 b(or)25 b(an)n(y)g(elemen)n(t)f
Fs(g)h Fp(2)f Fs(G)i Fu(one)f(can)g(write)94 3260 y Fs(g)h
Fu(=)d Fs(hg)336 3272 y Fn(0)400 3260 y Fu(for)k(some)f
Fs(h)d Fp(2)h Fs(H)7 b Fu(.)36 b(Then)28 b Fs(g)1280
3230 y Fm(\000)p Fn(1)1369 3260 y Fs(V)19 b(g)25 b Fu(=)e
Fs(g)1632 3225 y Fm(\000)p Fn(1)1629 3282 y(0)1721 3260
y Fs(h)1769 3230 y Fm(\000)p Fn(1)1858 3260 y Fs(V)18
b(hg)2012 3272 y Fn(0)2072 3260 y Fu(=)23 b Fs(g)2203
3225 y Fm(\000)p Fn(1)2200 3282 y(0)2292 3260 y Fs(V)18
b(g)2398 3272 y Fn(0)2458 3260 y Fp(\032)23 b Fs(V)c
Fu(.)p 3585 3211 57 4 v 3585 3260 4 50 v 3638 3260 V
3585 3263 57 4 v 94 3476 a Fv(Lemma)30 b(3.1.2)40 b Ft(If)30
b Fs(\036)23 b Fu(:)h Fs(G)f Fp(!)1085 3488 y Fn(2)1141
3455 y Fu(\026)1122 3476 y Fs(E)35 b Ft(is)30 b(a)g(c)l(o)l(cycle)h
(and)f Fs( )c Fu(=)d Fs(\036)-5 b Fp(j)24 b Fs(H)7 b
Ft(,)30 b(then)g Fs( )s Fu(\()p Fs(g)2525 3446 y Fn(2)2562
3476 y Fu(\))23 b Fp(2)g(f)p Fs(O)r(;)14 b(P)e Fp(g)30
b Ft(for)g(any)g Fs(g)c Fp(2)d Fs(G)p Fp(n)p Fs(H)7 b
Ft(.)94 3642 y(Pr)l(o)l(of)p Fu(.)39 b(Since)26 b Fs( )g
Fp(2)e Fs(H)799 3612 y Fn(1)836 3642 y Fu(\()p Fs(H)r(;)990
3654 y Fn(2)1046 3621 y Fu(\026)1027 3642 y Fs(E)5 b
Fu(\))1125 3612 y Fo(G=H)1274 3642 y Fu(,)28 b(it)e(follo)n(ws)f(that)i
Fs(g)1899 3654 y Fn(0)1936 3642 y Fs( )s Fu(\()p Fs(h)p
Fu(\))d(=)e Fs( )s Fu(\()p Fs(g)2348 3607 y Fm(\000)p
Fn(1)2345 3664 y(0)2437 3642 y Fs(hg)2525 3654 y Fn(0)2562
3642 y Fu(\))28 b(for)e(an)n(y)h Fs(h)c Fp(2)g Fs(H)7
b Fu(.)37 b(Substituting)94 3742 y Fs(h)23 b Fu(=)g Fs(g)296
3712 y Fn(2)361 3742 y Fu(w)n(e)k(get)1023 3839 y Fs(g)1063
3851 y Fn(0)1100 3839 y Fs( )s Fu(\()p Fs(g)1232 3809
y Fn(2)1269 3839 y Fu(\))92 b(=)g Fs( )s Fu(\()p Fs(g)1682
3804 y Fm(\000)p Fn(1)1679 3861 y(0)1771 3839 y Fs(g)1814
3809 y Fn(2)1851 3839 y Fs(g)1891 3851 y Fn(0)1928 3839
y Fu(\))23 b(=)g Fs( )s Fu(\()p Fs(g)2203 3804 y Fm(\000)p
Fn(1)2200 3861 y(0)2292 3839 y Fs(g)s Fu(\))18 b(+)g
Fs( )s Fu(\()p Fs(g)s(g)2640 3851 y Fn(0)2677 3839 y
Fu(\))1393 3942 y(=)92 b Fs( )s Fu(\()p Fs(g)s(g)1722
3954 y Fn(0)1759 3942 y Fu(\))19 b(+)f Fs( )s Fu(\()p
Fs(g)2025 3907 y Fm(\000)p Fn(1)2022 3964 y(0)2114 3942
y Fs(g)s Fu(\))1393 4045 y(=)92 b Fs( )s Fu(\()p Fs(g)s(g)1722
4057 y Fn(0)1759 4045 y Fs(g)1802 4009 y Fm(\000)p Fn(1)1799
4067 y(0)1891 4045 y Fs(g)s Fu(\))1393 4145 y(=)g Fs( )s
Fu(\()p Fs(g)1682 4114 y Fn(2)1719 4145 y Fu(\))1384
4244 y Fp(,)83 b Fs( )s Fu(\()p Fs(g)1682 4214 y Fn(2)1719
4244 y Fu(\))24 b Fp(2)f(f)p Fs(O)r(;)14 b(P)e Fp(g)23
b Fs(:)p 3585 4332 V 3585 4382 4 50 v 3638 4382 V 3585
4385 57 4 v 94 4664 a Fk(3.2)113 b Fj(F)-7 b(ormula)33
b(for)i(c)-5 b(o)g(cycles)94 4818 y Fu(No)n(w)28 b(let)f
Fs(g)444 4830 y Fn(0)509 4818 y Fu(b)r(e)h(some)e(elemen)n(t)g(in)h
Fs(G)p Fp(n)p Fs(H)7 b Fu(.)37 b(F)-7 b(or)27 b(an)n(y)g
Fs(Q)c Fp(2)1965 4830 y Fn(2)2021 4797 y Fu(\026)2002
4818 y Fs(E)28 b Fu(=)23 b Fp(f)p Fs(O)r(;)14 b(P)r(;)g(T)7
b(;)14 b(S)5 b Fp(g)27 b Fu(and)h(for)f(an)n(y)g(homomorphism)94
4917 y Fs( )g Fp(2)c Fs(H)7 b(om)p Fu(\()p Fs(H)r(;)596
4929 y Fn(2)653 4896 y Fu(\026)633 4917 y Fs(E)e Fu(\))28
b(denote)g(b)n(y)f Fs(\036)1191 4929 y Fo(Q; )1341 4917
y Fu(the)h(map)f(de\014ned)h(b)n(y)f(the)h(rule)1078
5154 y Fs(\036)1127 5166 y Fo(Q; )1249 5154 y Fu(\()p
Fs(g)s Fu(\))23 b(=)1467 5037 y Fi(\032)1571 5102 y Fs( )s
Fu(\()p Fs(g)s Fu(\))14 b Fs(;)459 b(g)25 b Fp(2)f Fs(H)1571
5205 y(g)1611 5217 y Fn(0)1648 5205 y Fs( )s Fu(\()p
Fs(g)1780 5169 y Fm(\000)p Fn(1)1777 5227 y(0)1869 5205
y Fs(g)s Fu(\))18 b(+)g Fs(Q)c(;)83 b(g)25 b Fp(2)f Fs(G)p
Fp(n)p Fs(H)29 b(:)3490 5154 y Fu(\(11\))1845 5697 y(8)p
eop
%%Page: 9 9
9 8 bop 94 98 a Fv(Lemma)30 b(3.2.1)40 b Ft(A)n(ny)k(c)l(o)l(cycle)i
Fs(\036)51 b Fu(:)f Fs(G)h Fp(!)1591 110 y Fn(2)1647
77 y Fu(\026)1628 98 y Fs(E)f Ft(is)45 b(of)h(the)f(form)g
Fs(\036)2373 110 y Fo(Q; )2541 98 y Ft(for)g(some)g Fs(Q)50
b Fp(2)3151 110 y Fn(2)3207 77 y Fu(\026)3188 98 y Fs(E)g
Ft(and)45 b Fs( )53 b Fp(2)94 197 y Fs(H)7 b(om)p Fu(\()p
Fs(H)r(;)438 209 y Fn(2)494 176 y Fu(\026)475 197 y Fs(E)e
Fu(\))p Ft(.)44 b(Conversely,)34 b(let)e Fs(Q)f Ft(and)h
Fs( )j Ft(b)l(e)c(as)h(ab)l(ove.)46 b(Then)32 b Fs(@)5
b(\036)2353 209 y Fo(Q; )2501 197 y Fu(=)26 b(0)31 b
Ft(if)i(and)f(only)g(if)h(the)e(fol)t(lowing)94 297 y(two)f(c)l
(onditions)h(hold:)219 397 y(\(i\))f Fs(g)382 409 y Fn(0)419
397 y Fs( )s Fu(\()p Fs(h)p Fu(\))23 b(=)g Fs( )s Fu(\()p
Fs(g)831 361 y Fm(\000)p Fn(1)828 419 y(0)920 397 y Fs(hg)1008
409 y Fn(0)1045 397 y Fu(\))30 b Ft(for)g(any)h Fs(h)22
b Fp(2)i Fs(H)7 b Ft(;)219 496 y(\(ii\))31 b Fs( )s Fu(\()p
Fs(g)500 466 y Fn(2)497 517 y(0)537 496 y Fu(\))23 b(=)g
Fs(P)41 b Ft(for)31 b Fs(Q)23 b Fp(2)g(f)p Fs(T)7 b(;)14
b(S)5 b Fp(g)28 b Ft(and)i Fs( )s Fu(\()p Fs(g)1628 466
y Fn(2)1625 517 y(0)1665 496 y Fu(\))23 b(=)g Fs(O)32
b Ft(for)f Fs(Q)23 b Fp(2)g(f)p Fs(O)r(;)14 b(P)e Fp(g)p
Ft(.)94 667 y(Pr)l(o)l(of)p Fu(.)39 b(If)28 b Fs(\036)g
Fu(is)e(a)h(1-co)r(cycle)f(with)h(co)r(e\016cien)n(ts)f(in)1736
679 y Fn(2)1792 646 y Fu(\026)1773 667 y Fs(E)33 b Fu(then)28
b(for)f(an)n(y)g Fs(g)e Fp(2)f Fs(G)k Fu(w)n(e)f(ha)n(v)n(e)1052
838 y Fs(@)5 b(\036)p Fu(\()p Fs(g)1222 850 y Fn(0)1259
838 y Fs(;)14 b(g)1339 802 y Fm(\000)p Fn(1)1336 860
y(0)1427 838 y Fs(g)s Fu(\))23 b(=)g Fs(g)1653 850 y
Fn(0)1690 838 y Fs(\036)p Fu(\()p Fs(g)1814 802 y Fm(\000)p
Fn(1)1811 860 y(0)1903 838 y Fs(g)s Fu(\))c(+)f Fs(\036)p
Fu(\()p Fs(g)s Fu(\))h(+)f Fs(\036)p Fu(\()p Fs(g)2459
850 y Fn(0)2496 838 y Fu(\))24 b(=)e(0)94 1008 y(for)28
b(some)e Fs(g)470 1020 y Fn(0)529 1008 y Fp(2)e Fs(G)p
Fp(n)p Fs(H)7 b Fu(.)36 b(Therefore)27 b Fs(\036)c Fu(=)g
Fs(\036)1436 1020 y Fo(Q; )1559 1008 y Fu(,)k(where)g
Fs(Q)c Fu(=)g Fs(\036)p Fu(\()p Fs(g)2147 1020 y Fn(0)2184
1008 y Fu(\))28 b(and)g Fs( )e Fu(=)c Fs(\036)-5 b Fp(j)2640
1020 y Fo(H)2704 1008 y Fu(.)219 1108 y(If)28 b Fs(h)350
1120 y Fn(1)387 1108 y Fs(;)14 b(h)472 1120 y Fn(2)532
1108 y Fp(2)24 b Fs(H)7 b Fu(,)27 b(then)h(it)f(is)g(clear)e(that)j
Fs(@)5 b(\036)1569 1120 y Fo(Q; )1691 1108 y Fu(\()p
Fs(h)1771 1120 y Fn(1)1809 1108 y Fs(;)14 b(h)1894 1120
y Fn(2)1931 1108 y Fu(\))23 b(=)g(0.)36 b(Let)28 b Fs(g)e
Fp(2)d Fs(G)p Fp(n)p Fs(H)34 b Fu(and)28 b Fs(h)23 b
Fp(2)g Fs(H)7 b Fu(.)37 b(Then)773 1284 y Fs(@)5 b(\036)871
1296 y Fo(Q; )994 1284 y Fu(\()p Fs(g)s(;)14 b(h)p Fu(\))83
b(=)f Fs(g)s( )s Fu(\()p Fs(h)p Fu(\))19 b(+)f Fs(g)1770
1296 y Fn(0)1807 1284 y Fs( )s Fu(\()p Fs(g)1939 1249
y Fm(\000)p Fn(1)1936 1306 y(0)2028 1284 y Fs(g)s(h)p
Fu(\))g(+)g Fs(Q)g Fu(+)g Fs(g)2459 1296 y Fn(0)2496
1284 y Fs( )s Fu(\()p Fs(g)2628 1249 y Fm(\000)p Fn(1)2625
1306 y(0)2717 1284 y Fs(g)s Fu(\))g(+)g Fs(Q)1269 1387
y Fu(=)82 b Fs(g)s( )s Fu(\()p Fs(h)p Fu(\))19 b(+)f
Fs(g)1770 1399 y Fn(0)1807 1387 y Fs( )s Fu(\()p Fs(g)1939
1351 y Fm(\000)p Fn(1)1936 1409 y(0)2028 1387 y Fs(g)s
Fu(\))g(+)g Fs(g)2244 1399 y Fn(0)2281 1387 y Fs( )s
Fu(\()p Fs(h)p Fu(\))h(+)f Fs(g)2592 1399 y Fn(0)2629
1387 y Fs( )s Fu(\()p Fs(g)2761 1351 y Fm(\000)p Fn(1)2758
1409 y(0)2850 1387 y Fs(g)s Fu(\))1269 1487 y(=)82 b
Fs(g)s( )s Fu(\()p Fs(h)p Fu(\))19 b(+)f Fs(g)1770 1499
y Fn(0)1807 1487 y Fs( )s Fu(\()p Fs(h)p Fu(\))1269 1637
y(=)1416 1520 y Fi(\032)1520 1586 y Fu(2)p Fs( )s Fu(\()p
Fs(h)p Fu(\))c Fs(;)187 b( )s Fu(\()p Fs(h)p Fu(\))24
b Fp(2)f(f)p Fs(O)r(;)14 b(P)e Fp(g)1520 1686 y Fu(2)p
Fs(g)1602 1698 y Fn(0)1638 1686 y Fs( )s Fu(\()p Fs(h)p
Fu(\))i Fs(;)111 b( )s Fu(\()p Fs(h)p Fu(\))24 b Fp(2)f(f)p
Fs(T)7 b(;)14 b(S)5 b Fp(g)1269 1785 y Fu(=)82 b(0)94
1952 y(Note)28 b(that)g(w)n(e)f(replace)f(signs)g(")p
Fp(\000)p Fu(")g(b)n(y)h("+")g(b)r(ecause)g Fs(\036)1905
1964 y Fo(Q; )2055 1952 y Fu(is)g(a)g(map)f(in)n(to)h(a)g(2-torsion)d
(group.)219 2052 y(If)k Fs(h)23 b Fp(2)g Fs(H)r(;)14
b(g)26 b Fp(2)d Fs(G)p Fp(n)p Fs(H)7 b Fu(,)28 b(then)g(for)f(an)n(y)g
Fs(Q)g Fu(and)g Fs( )k Fu(w)n(e)c(ha)n(v)n(e)708 2225
y Fs(@)5 b(\036)806 2237 y Fo(Q; )928 2225 y Fu(\()p
Fs(h;)14 b(g)s Fu(\))83 b(=)f Fs(g)1390 2237 y Fn(0)1427
2225 y Fs( )s Fu(\()p Fs(g)1559 2190 y Fm(\000)p Fn(1)1556
2247 y(0)1648 2225 y Fs(g)s Fu(\))19 b(+)f Fs(Q)g Fu(+)g
Fs(g)2032 2237 y Fn(0)2069 2225 y Fs( )s Fu(\()p Fs(g)2201
2190 y Fm(\000)p Fn(1)2198 2247 y(0)2290 2225 y Fs(hg)s
Fu(\))g(+)g Fs(Q)g Fu(+)g Fs( )s Fu(\()p Fs(h)p Fu(\))1203
2328 y(=)82 b Fs(g)1390 2340 y Fn(0)1427 2328 y Fs( )s
Fu(\()p Fs(g)1559 2292 y Fm(\000)p Fn(1)1556 2350 y(0)1648
2328 y Fs(g)s Fu(\))19 b(+)f Fs(g)1865 2340 y Fn(0)1901
2328 y Fs( )s Fu(\()p Fs(g)2033 2292 y Fm(\000)p Fn(1)2030
2350 y(0)2123 2328 y Fs(hg)2211 2340 y Fn(0)2247 2328
y Fu(\))h(+)f Fs(g)2421 2340 y Fn(0)2458 2328 y Fs( )s
Fu(\()p Fs(g)2590 2292 y Fm(\000)p Fn(1)2587 2350 y(0)2679
2328 y Fs(g)s Fu(\))g(+)g Fs( )s Fu(\()p Fs(h)p Fu(\))1203
2431 y(=)82 b Fs(g)1390 2443 y Fn(0)1427 2431 y Fs( )s
Fu(\()p Fs(g)1559 2395 y Fm(\000)p Fn(1)1556 2453 y(0)1648
2431 y Fs(hg)1736 2443 y Fn(0)1773 2431 y Fu(\))19 b(+)f
Fs( )s Fu(\()p Fs(h)p Fu(\))23 b Fs(:)94 2613 y Fu(Therefore,)k(if)g
Fs(@)5 b(\036)668 2625 y Fo(Q; )813 2613 y Fu(=)23 b(0,)k(then)h
Fs(g)1222 2625 y Fn(0)1259 2613 y Fs( )s Fu(\()p Fs(g)1391
2577 y Fm(\000)p Fn(1)1388 2635 y(0)1480 2613 y Fs(hg)1568
2625 y Fn(0)1605 2613 y Fu(\))19 b(+)f Fs( )s Fu(\()p
Fs(h)p Fu(\))23 b(=)g(0,)k(that)h(is)1476 2784 y Fs(g)1516
2796 y Fn(0)1552 2784 y Fs( )s Fu(\()p Fs(h)p Fu(\))c(=)f
Fs( )s Fu(\()p Fs(g)1965 2748 y Fm(\000)p Fn(1)1962 2806
y(0)2054 2784 y Fs(hg)2142 2796 y Fn(0)2178 2784 y Fu(\))h
Fs(:)1233 b Fu(\(12\))219 2954 y(If)28 b Fs(g)342 2966
y Fn(1)402 2954 y Fp(2)23 b Fs(G)p Fp(n)p Fs(H)r(;)14
b(g)735 2966 y Fn(2)795 2954 y Fp(2)24 b Fs(G)p Fp(n)p
Fs(H)7 b Fu(,)27 b(then)h(for)f(an)n(y)g Fs(Q)g Fu(and)h
Fs( )i Fu(w)n(e)d(ha)n(v)n(e:)564 3127 y Fs(@)5 b(\036)662
3139 y Fo(Q; )785 3127 y Fu(\()p Fs(g)857 3139 y Fn(1)894
3127 y Fs(;)14 b(g)971 3139 y Fn(2)1008 3127 y Fu(\))110
b(=)f Fs(g)1364 3139 y Fn(1)1401 3127 y Fu(\()p Fs(g)1473
3139 y Fn(0)1510 3127 y Fs( )s Fu(\()p Fs(g)1642 3092
y Fm(\000)p Fn(1)1639 3150 y(0)1731 3127 y Fs(g)1771
3139 y Fn(2)1808 3127 y Fu(\))19 b(+)f Fs(Q)p Fu(\))g(+)g
Fs( )s Fu(\()p Fs(g)2270 3139 y Fn(1)2307 3127 y Fs(g)2347
3139 y Fn(2)2384 3127 y Fu(\))h(+)f Fs(g)2558 3139 y
Fn(0)2595 3127 y Fs( )s Fu(\()p Fs(g)2727 3092 y Fm(\000)p
Fn(1)2724 3150 y(0)2816 3127 y Fs(g)2856 3139 y Fn(1)2893
3127 y Fu(\))g(+)g Fs(Q)1150 3230 y Fu(=)109 b Fs( )s
Fu(\()p Fs(g)1456 3195 y Fm(\000)p Fn(1)1453 3252 y(0)1545
3230 y Fs(g)1585 3242 y Fn(2)1622 3230 y Fu(\))19 b(+)f
Fs( )s Fu(\()p Fs(g)1885 3242 y Fn(1)1922 3230 y Fs(g)1962
3242 y Fn(2)1999 3230 y Fu(\))g(+)g Fs(g)2172 3242 y
Fn(0)2209 3230 y Fs( )s Fu(\()p Fs(g)2341 3195 y Fm(\000)p
Fn(1)2338 3252 y(0)2430 3230 y Fs(g)2470 3242 y Fn(1)2507
3230 y Fu(\))h(+)f Fs(g)2681 3242 y Fn(1)2718 3230 y
Fs(Q)g Fu(+)g Fs(Q)1123 3312 y Fn(\(12\))1150 3366 y
Fu(=)109 b Fs( )s Fu(\()p Fs(g)1456 3331 y Fm(\000)p
Fn(1)1453 3388 y(0)1545 3366 y Fs(g)1585 3378 y Fn(2)1622
3366 y Fu(\))19 b(+)f Fs( )s Fu(\()p Fs(g)1885 3378 y
Fn(1)1922 3366 y Fs(g)1962 3378 y Fn(2)1999 3366 y Fu(\))g(+)g
Fs( )s Fu(\()p Fs(g)2264 3331 y Fm(\000)p Fn(1)2261 3388
y(0)2354 3366 y Fs(g)2397 3331 y Fm(\000)p Fn(1)2394
3388 y(0)2485 3366 y Fs(g)2525 3378 y Fn(1)2562 3366
y Fs(g)2602 3378 y Fn(0)2639 3366 y Fu(\))h(+)f Fs(g)2813
3378 y Fn(1)2850 3366 y Fs(Q)g Fu(+)g Fs(Q)1150 3469
y Fu(=)109 b Fs( )s Fu(\()p Fs(g)1456 3434 y Fm(\000)p
Fn(1)1453 3491 y(0)1545 3469 y Fs(g)1585 3481 y Fn(2)1622
3469 y Fu(\))19 b(+)f Fs( )s Fu(\()p Fs(g)1885 3481 y
Fn(1)1922 3469 y Fs(g)1962 3481 y Fn(2)1999 3469 y Fu(\))g(+)g
Fs( )s Fu(\()p Fs(g)2264 3439 y Fn(2)2261 3490 y(0)2302
3469 y Fu(\))g(+)g Fs( )s Fu(\()p Fs(g)2564 3481 y Fn(1)2602
3469 y Fs(g)2642 3481 y Fn(0)2678 3469 y Fu(\))h(+)f
Fs(g)2852 3481 y Fn(1)2889 3469 y Fs(Q)g Fu(+)g Fs(Q)23
b(:)94 3636 y Fu(Observ)n(e)j(that)667 3788 y Fs( )s
Fu(\()p Fs(g)799 3753 y Fm(\000)p Fn(1)796 3810 y(0)888
3788 y Fs(g)928 3800 y Fn(2)965 3788 y Fu(\))19 b(+)f
Fs( )s Fu(\()p Fs(g)1228 3800 y Fn(1)1265 3788 y Fs(g)1305
3800 y Fn(2)1342 3788 y Fu(\))g(+)h Fs( )s Fu(\()p Fs(g)1605
3800 y Fn(1)1642 3788 y Fs(g)1682 3800 y Fn(0)1719 3788
y Fu(\))83 b(=)f Fs( )s Fu(\()p Fs(g)2110 3800 y Fn(1)2148
3788 y Fs(g)2188 3800 y Fn(0)2224 3788 y Fu(\))19 b(+)f
Fs( )s Fu(\()p Fs(g)2490 3753 y Fm(\000)p Fn(1)2487 3810
y(0)2579 3788 y Fs(g)2619 3800 y Fn(2)2656 3788 y Fu(\))h(+)f
Fs( )s Fu(\()p Fs(g)2919 3800 y Fn(1)2956 3788 y Fs(g)2996
3800 y Fn(2)3033 3788 y Fu(\))1834 3891 y(=)82 b Fs( )s
Fu(\()p Fs(g)2110 3903 y Fn(1)2148 3891 y Fs(g)2188 3903
y Fn(0)2224 3891 y Fs(g)2267 3856 y Fm(\000)p Fn(1)2264
3913 y(0)2356 3891 y Fs(g)2396 3903 y Fn(2)2433 3891
y Fu(\))19 b(+)f Fs( )s Fu(\()p Fs(g)2696 3903 y Fn(1)2733
3891 y Fs(g)2773 3903 y Fn(2)2810 3891 y Fu(\))1834 3991
y(=)82 b Fs( )s Fu(\()p Fs(g)2110 4003 y Fn(1)2148 3991
y Fs(g)2188 4003 y Fn(2)2224 3991 y Fu(\))19 b(+)f Fs( )s
Fu(\()p Fs(g)2487 4003 y Fn(1)2524 3991 y Fs(g)2564 4003
y Fn(2)2601 3991 y Fu(\))1834 4090 y(=)82 b(2)p Fs( )s
Fu(\()p Fs(g)2152 4102 y Fn(1)2189 4090 y Fs(g)2229 4102
y Fn(2)2266 4090 y Fu(\))1834 4190 y(=)g(0)94 4356 y(Hence)28
b(one)f(can)h(con)n(tin)n(ue:)614 4508 y Fs(@)5 b(\036)712
4520 y Fo(Q; )835 4508 y Fu(\()p Fs(g)907 4520 y Fn(1)944
4508 y Fs(;)14 b(g)1021 4520 y Fn(2)1057 4508 y Fu(\))84
b(=)e Fs( )s Fu(\()p Fs(g)1452 4473 y Fm(\000)p Fn(1)1449
4531 y(0)1541 4508 y Fs(g)1581 4520 y Fn(2)1618 4508
y Fu(\))19 b(+)f Fs( )s Fu(\()p Fs(g)1881 4520 y Fn(1)1918
4508 y Fs(g)1958 4520 y Fn(2)1995 4508 y Fu(\))h(+)f
Fs( )s Fu(\()p Fs(g)2261 4478 y Fn(2)2258 4529 y(0)2298
4508 y Fu(\))h(+)f Fs( )s Fu(\()p Fs(g)2561 4520 y Fn(1)2598
4508 y Fs(g)2638 4520 y Fn(0)2675 4508 y Fu(\))g(+)g
Fs(g)2848 4520 y Fn(1)2885 4508 y Fs(Q)g Fu(+)g Fs(Q)1173
4608 y Fu(=)82 b Fs( )s Fu(\()p Fs(g)1452 4578 y Fn(2)1449
4629 y(0)1489 4608 y Fu(\))19 b(+)f Fs(g)1663 4620 y
Fn(1)1700 4608 y Fs(Q)g Fu(+)g Fs(Q)1173 4708 y Fu(=)82
b Fs( )s Fu(\()p Fs(g)1452 4678 y Fn(2)1449 4728 y(0)1489
4708 y Fu(\))19 b(+)f Fs(g)1663 4720 y Fn(0)1700 4708
y Fs(Q)g Fu(+)g Fs(Q)23 b(:)94 4875 y Fu(Therefore,)k(if)g
Fs(@)5 b(\036)668 4887 y Fo(Q; )813 4875 y Fu(=)23 b(0,)k(then)1532
4974 y Fs( )s Fu(\()p Fs(g)1664 4940 y Fn(2)1661 4995
y(0)1701 4974 y Fu(\))d(=)e Fs(g)1884 4986 y Fn(0)1921
4974 y Fs(Q)c Fu(+)g Fs(Q)23 b(:)p 3585 5067 57 4 v 3585
5116 4 50 v 3638 5116 V 3585 5119 57 4 v 94 5332 a Fv(Remark)31
b(3.2.1)40 b Ft(Cle)l(arly,)g(the)c(c)l(ondition)i Fu(\()p
Fs(i)p Fu(\))e Ft(is)h(e)l(quivalent)g(to)f(the)g(c)l(ondition)i
Fs( )g Fp(2)d Fu(\()p Fs(H)7 b(om)p Fu(\()p Fs(H)r(;)3321
5344 y Fn(2)3378 5311 y Fu(\026)3358 5332 y Fs(E)f Fu(\)\))3489
5302 y Fo(G=H)94 5432 y Ft(and)31 b(c)l(an)e(b)l(e)h(de)l(duc)l(e)l(d)h
(dir)l(e)l(ctly)f(fr)l(om)h(the)e(isomorphism)k(10.)1845
5697 y Fu(9)p eop
%%Page: 10 10
10 9 bop 94 98 a Fk(3.3)113 b Fj(Classi\014c)-5 b(ation)33
b(of)i(c)-5 b(o)g(cycles)94 251 y Fu(No)n(w)29 b(one)g(can)h(classify)c
(all)h(1-co)r(cycles)g Fs(\036)g Fu(=)e Fs(\036)1604
263 y Fo(Q; )1757 251 y Fu(of)k(the)h(group)e Fs(G)i
Fu(with)f(co)r(e\016cien)n(ts)f(in)3055 263 y Fn(2)3112
230 y Fu(\026)3093 251 y Fs(E)34 b Fu(according)27 b(to)94
351 y(the)h(n)n(um)n(b)r(er)f(of)g(elemen)n(ts)f(in)h(im)n(\()p
Fs( )s Fu(\).)219 450 y(Denote)d(b)n(y)f Fs(\030)28 b
Fu(the)c(surjectiv)n(e)e(homomorphism)c Fs(G)24 b Fp(!)f(f)p
Fs(O)r(;)14 b(P)e Fp(g)23 b Fu(with)g(k)n(ernel)f Fs(H)7
b Fu(.)35 b(Then)24 b Fs(\030)j Fu(=)c Fs(@)5 b(T)33
b Fu(=)23 b Fs(@)5 b(S)28 b Fu(and)94 550 y Fs(\030)36
b Fu(is)30 b(a)h(unique)f(non-trivial)d(1-cob)r(oundary)i(of)j
Fs(G)f Fu(with)g(co)r(e\016cien)n(ts)f(in)2462 562 y
Fn(2)2519 529 y Fu(\026)2499 550 y Fs(E)5 b Fu(.)48 b(If)32
b(#)14 b(im)n(\()p Fs( )s Fu(\))30 b(=)f(1,)i(i.e.)47
b Fs( )33 b Fu(=)28 b(0,)94 650 y(then)j(the)f(form)n(ula)d(11)i(and)g
(Lemma)f(3.2.1)g(sho)n(w)h(that)h(either)f Fs(\036)e
Fu(=)f(0)j(or)g Fs(\036)e Fu(=)f Fs(\030)t Fu(.)44 b(In)30
b(b)r(oth)g(cases)f(cls)n(\()p Fs(\036)p Fu(\))f(=)e(0)94
749 y(in)h Fs(H)267 719 y Fn(1)304 749 y Fu(\()p Fs(G;)453
761 y Fn(2)509 728 y Fu(\026)490 749 y Fs(E)5 b Fu(\).)219
849 y(Let)39 b Fs(\036)428 861 y Fo(Q; )589 849 y Fu(b)r(e)h(a)e(co)r
(cycle.)69 b(Let)39 b Fs( )1360 861 y Fo(T)1451 849 y
Fu(and)f Fs( )1677 861 y Fo(S)1764 849 y Fu(are)g(the)h(comp)r
(ositions)c(of)k Fs( )j Fu(with)c(the)h(pro)5 b(jections)37
b(of)94 948 y(the)f(group)501 960 y Fn(2)557 927 y Fu(\026)538
948 y Fs(E)k Fu(on)34 b Fp(f)p Fs(O)r(;)14 b(T)e Fp(g)34
b Fu(and)g Fp(f)p Fs(O)r(;)14 b(S)5 b Fp(g)34 b Fu(resp)r(ectiv)n(ely)
-7 b(.)56 b(Let)34 b Fs(U)44 b Fu(=)34 b(k)n(er)o(\()p
Fs( )2538 960 y Fo(T)2591 948 y Fu(\).)59 b(Then)34 b(k)n(er)o(\()p
Fs( )3125 960 y Fo(S)3174 948 y Fu(\))h(=)f Fs(g)3383
913 y Fm(\000)p Fn(1)3380 971 y(0)3472 948 y Fs(U)9 b(g)3578
960 y Fn(0)3615 948 y Fu(.)94 1048 y(The)33 b(condition)c(#)14
b(im)o(\()p Fs( )s Fu(\))31 b(=)f(2)i(is)f(equiv)-5 b(alen)n(t)30
b(to)h(the)i(condition)d Fs(g)2287 1013 y Fm(\000)p Fn(1)2284
1070 y(0)2376 1048 y Fs(U)9 b(g)2482 1060 y Fn(0)2549
1048 y Fu(=)30 b Fs(U)9 b Fu(,)33 b Fs(g)2806 1060 y
Fn(0)2873 1048 y Fp(2)e Fs(G)p Fp(n)p Fs(H)7 b Fu(.)50
b(In)32 b(this)g(case)94 1148 y Fs( )39 b Fu(is)34 b(a)h(surjectiv)n(e)
e(homomorphism)e Fs(H)42 b Fp(!)36 b(f)p Fs(O)r(;)14
b(P)e Fp(g)35 b Fu(with)f(k)n(ernel)g Fs(U)9 b Fu(.)59
b(The)36 b(group)e Fs(U)44 b Fu(is)34 b(normal)e(in)j
Fs(G)g Fu(b)n(y)94 1247 y(Lemma)d(3.1.1.)52 b(Consider)31
b(the)j(quotien)n(t)e(group)g Fs(G=U)9 b Fu(.)54 b(There)33
b(are)f(t)n(w)n(o)g(p)r(ossible)f(cases)h(for)h Fs(G=U)9
b Fu(:)48 b(either)94 1347 y Fs(G=U)295 1325 y Fp(\030)295
1351 y Fu(=)387 1347 y Fl(Z)o Fs(=)p Fu(2)14 b Fp(\010)20
b Fl(Z)o Fs(=)p Fu(2)k(or)29 b Fs(G=U)1104 1325 y Fp(\030)1104
1351 y Fu(=)1197 1347 y Fl(Z)o Fs(=)p Fu(4)o(.)39 b(In)31
b(the)g(\014rst)f(case)g(one)g(can)g(c)n(ho)r(ose)f Fs(g)2625
1359 y Fn(0)2689 1347 y Fp(2)g Fs(G)p Fp(n)p Fs(H)37
b Fu(suc)n(h)30 b(that)g Fs(g)3401 1317 y Fn(2)3398 1368
y(0)3466 1347 y Fp(2)e Fs(U)9 b Fu(.)94 1447 y(By)33
b(Lemma)e(3.2.1)h(the)h(p)r(oin)n(t)g Fs(Q)f Fu(lie)g(in)g
Fp(f)p Fs(O)r(;)14 b(P)e Fp(g)p Fu(.)53 b(But)33 b Fs(\036)1970
1459 y Fo(P)r(; )2106 1447 y Fu(+)22 b Fs(\036)2242 1459
y Fo(O)r(; )2397 1447 y Fu(=)32 b Fs(\030)k Fu(=)c Fs(@)5
b(T)12 b Fu(.)52 b(Consequen)n(tly)-7 b(,)33 b(w)n(e)f(can)94
1546 y(consider)c(only)f(co)r(cycles)h Fs(\036)973 1558
y Fo(O)r(; )1095 1546 y Fu(.)42 b(If)30 b Fs(G=U)1443
1524 y Fp(\030)1443 1550 y Fu(=)1534 1546 y Fv(Z)p Fs(=)p
Fv(4)e Fu(then)i Fs(g)1944 1516 y Fn(2)1941 1567 y(0)2007
1546 y Fp(2)c Fs(H)7 b Fp(n)p Fs(U)i Fu(.)41 b(By)29
b(Lemma)e(3.2.1)h(the)h(p)r(oin)n(t)g Fs(Q)g Fu(lies)e(in)94
1646 y Fp(f)p Fs(T)7 b(;)14 b(S)5 b Fp(g)p Fu(.)35 b(Adding)27
b Fs(\030)32 b Fu(one)27 b(can)h(consider)d(only)h(co)r(cycles)g
Fs(\036)1924 1658 y Fo(T)5 b(; )2039 1646 y Fu(.)94 1806
y Fv(De\014nition)31 b(3.3.1)40 b Ft(L)l(et)g Fs(\036)j
Fu(=)f Fs(\036)1180 1818 y Fo(Q; )1343 1806 y Ft(b)l(e)e(a)h(c)l(o)l
(cycle)h(with)f Fu(#)14 b(im)n(\()p Fs( )s Fu(\))43 b(=)f(2)e
Ft(and)g Fs(U)52 b Fu(=)41 b(k)n(er)o(\()p Fs( )s Fu(\))p
Ft(.)72 b(If)40 b Fs(G=U)3573 1783 y Fp(\030)3573 1810
y Fu(=)94 1905 y Fl(Z)p Fs(=)o Fu(2)14 b Fp(\010)19 b
Fl(Z)o Fs(=)p Fu(2)25 b Ft(and)32 b Fs(Q)26 b Fu(=)f
Fs(O)r Ft(,)33 b(then)f(we)g(say)g(that)f Fs(\036)h Ft(is)g(a)g(c)l(o)l
(cycle)h(of)f(the)g(typ)l(e)f Fs(I)7 b Ft(.)44 b(If)32
b Fs(G=U)2946 1883 y Fp(\030)2946 1909 y Fu(=)3037 1905
y Fl(Z)o Fs(=)p Fu(4)25 b Ft(and)32 b Fs(Q)25 b Fu(=)h
Fs(T)12 b Ft(,)94 2005 y(then)30 b(we)g(say)h(that)e
Fs(\036)h Ft(is)h(a)f(c)l(o)l(cycle)h(of)f(the)g(typ)l(e)g
Fs(I)7 b(I)g Ft(.)219 2164 y Fu(No)n(w)35 b(let)g(us)h(consider)e(the)i
(case)f(of)h(a)f(co)r(cycle)f Fs(\036)j Fu(=)f Fs(\036)2036
2176 y Fo(Q; )2195 2164 y Fu(with)f(#)14 b(im)n(\()p
Fs( )s Fu(\))38 b(=)e(4.)61 b(The)36 b(corresp)r(onding)94
2264 y(homomorphism)d(can)k(b)r(e)h(uniquely)d(determined)h(b)n(y)h(a)g
(subgroup)g Fs(U)46 b Fu(of)37 b(index)g(2)g(in)g Fs(H)44
b Fu(suc)n(h)37 b(that)h Fs(U)48 b Fp(6)p Fu(=)94 2364
y Fs(g)137 2328 y Fm(\000)p Fn(1)134 2386 y(0)226 2364
y Fs(U)9 b(g)332 2376 y Fn(0)401 2364 y Fu(for)31 b(some)f
Fs(g)784 2376 y Fn(0)851 2364 y Fp(2)h Fs(G)p Fp(n)p
Fs(H)38 b Fu(\(note)32 b(that)h Fs(g)1600 2328 y Fm(\000)p
Fn(1)1597 2386 y(0)1688 2364 y Fs(U)9 b(g)1794 2376 y
Fn(0)1861 2364 y Fu(=)30 b Fs(g)1999 2334 y Fm(0)1996
2384 y Fn(0)2033 2322 y Fm(\000)p Fn(1)2122 2364 y Fs(U)9
b(g)2231 2334 y Fm(0)2228 2384 y Fn(0)2296 2364 y Fu(for)32
b(an)n(y)f(t)n(w)n(o)g(elemen)n(ts)f Fs(g)3134 2376 y
Fn(0)3170 2364 y Fs(;)14 b(g)3250 2334 y Fm(0)3247 2384
y Fn(0)3314 2364 y Fp(2)31 b Fs(G)p Fp(n)p Fs(H)7 b Fu(\).)94
2463 y(Let)37 b Fs(U)309 2475 y Fo(O)404 2463 y Fu(=)h
Fs(U)33 b Fp(\\)25 b Fs(g)720 2428 y Fm(\000)p Fn(1)717
2486 y(0)809 2463 y Fs(U)9 b(g)915 2475 y Fn(0)952 2463
y Fu(.)64 b(The)37 b(condition)e Fs(U)47 b Fp(6)p Fu(=)38
b Fs(g)1843 2428 y Fm(\000)p Fn(1)1840 2486 y(0)1932
2463 y Fs(U)9 b(g)2038 2475 y Fn(0)2111 2463 y Fu(implies)33
b(that)38 b(the)f(quotien)n(t)e(group)h Fs(G=U)3489 2475
y Fo(O)3582 2463 y Fu(is)94 2563 y(noncomm)n(utativ)n(e)d(of)j(order)f
(8.)62 b(Moreo)n(v)n(er,)36 b Fs(G=U)1752 2575 y Fo(O)1844
2563 y Fu(has)g(3)g(elemen)n(ts)e(of)i(order)f(2)h(lying)e(in)h(the)i
(subgroup)94 2663 y Fs(H)r(=U)264 2675 y Fo(O)320 2663
y Fu(.)44 b(Therefore)28 b Fs(G=U)929 2675 y Fo(O)1015
2663 y Fu(is)h(isomorphic)d(to)j(the)h(dihedral)e(group)g
Fs(D)2404 2675 y Fn(4)2471 2663 y Fu(\(see,)j(for)e(example,)f([Hall59)
m(],)i(4.4\).)94 2762 y(But)25 b Fs(D)325 2774 y Fn(4)386
2762 y Fu(has)e(\014v)n(e)h(elemen)n(ts)e(of)i(order)e(2.)36
b(Consequen)n(tly)-7 b(,)23 b(there)h(exists)e(an)i(elemen)n(t)e
Fs(g)2837 2774 y Fn(0)2898 2762 y Fu(in)h Fs(G)p Fp(n)p
Fs(H)31 b Fu(suc)n(h)24 b(that)g Fs(g)3601 2732 y Fn(2)3598
2783 y(0)94 2862 y Fu(lies)j(in)h Fs(U)394 2874 y Fo(O)450
2862 y Fu(.)41 b(By)28 b(Lemma)f(3.2.1)g(the)i(p)r(oin)n(t)f
Fs(Q)h Fu(lies)d Fp(f)p Fs(O)r(;)14 b(P)e Fp(g)p Fu(.)40
b(Adding)28 b Fs(\030)33 b Fu(if)28 b(it)h(is)e(necessary)g(one)i(can)f
(consider)94 2962 y(only)f(the)h(case)e Fs(Q)d Fu(=)g
Fs(O)r Fu(.)94 3121 y Fv(De\014nition)31 b(3.3.2)40 b
Ft(Co)l(cycles)31 b Fs(\036)24 b Fu(=)f Fs(\036)1325
3133 y Fo(T)5 b(; )1469 3121 y Ft(with)31 b Fu(#)14 b(im)n(\()p
Fs( )s Fu(\))24 b(=)e(4)29 b Ft(wil)t(l)i(b)l(e)f(c)l(al)t(le)l(d)h(c)l
(o)l(cycles)g(of)g(typ)l(e)f Fs(I)7 b(I)g(I)g Ft(.)219
3281 y Fu(In)28 b(fact)g(w)n(e)f(pro)n(v)n(ed)f(the)94
3441 y Fv(Prop)s(osition)31 b(3.3.1)39 b Ft(A)n(ny)27
b(element)h(in)1441 3453 y Fn(2)1478 3441 y Fs(H)1554
3410 y Fn(1)1591 3441 y Fu(\()p Fs(G;)1745 3420 y Fu(\026)1725
3441 y Fs(E)6 b Fu(\))28 b Ft(c)l(an)f(b)l(e)h(r)l(epr)l(esente)l(d)f
(by)h(c)l(o)l(cycle)h(of)f(one)g(of)g(thr)l(e)l(e)g(typ)l(es)94
3540 y Fs(I)7 b Ft(,)31 b Fs(I)7 b(I)36 b Ft(and)31 b
Fs(I)7 b(I)g(I)g Ft(.)94 3771 y Fk(3.4)113 b Fj(Dir)-5
b(e)g(ct)34 b(c)-5 b(alculations)34 b(of)h(c)-5 b(o)g(cycles)94
3924 y Fv(Prop)s(osition)31 b(3.4.1)39 b Ft(L)l(et)c
Fs(\036)f Fu(=)e Fs(\036)1226 3936 y Fo(Q; )1384 3924
y Ft(b)l(e)k(a)f(c)l(o)l(cycle)i(of)f(the)f(gr)l(oup)h
Fs(G)f Ft(with)h(c)l(o)l(e\016cients)g(in)3169 3936 y
Fn(2)3226 3904 y Fu(\026)3206 3924 y Fs(E)41 b Ft(and)35
b Fs(U)42 b Fu(=)94 4024 y(k)n(er)o(\()p Fs( )s Fu(\))p
Ft(.)d(If)27 b Fs(\036)h Ft(is)f(a)g(c)l(o)l(cycle)i(of)f(typ)l(e)f
Fs(I)7 b Ft(,)28 b(then)f Fs(\036)g Ft(is)h(the)f(surje)l(ctive)g
(homomorphism)j Fs(G)23 b Fp(!)g(f)p Fs(O)r(;)14 b(P)e
Fp(g)27 b Ft(with)g(kernel)94 4124 y Fs(U)h Fp([)18 b
Fs(g)292 4136 y Fn(0)329 4124 y Fs(U)9 b Ft(.)39 b(If)30
b Fs(\036)23 b Fu(=)g Fs(\036)755 4136 y Fo(T)5 b(; )900
4124 y Ft(has)30 b(typ)l(e)g Fs(I)7 b(I)g Ft(,)30 b(then)1159
4451 y Fs(\036)1208 4463 y Fo(T)5 b(; )1323 4451 y Fu(\()p
Fs(g)s Fu(\))23 b(=)1540 4231 y Fi(8)1540 4306 y(>)1540
4331 y(>)1540 4355 y(<)1540 4505 y(>)1540 4530 y(>)1540
4555 y(:)1656 4301 y Fs(O)i(;)51 b Ft(if)38 b Fs(g)26
b Fp(2)d Fs(U)1656 4400 y(T)34 b(;)51 b Ft(if)37 b Fs(g)26
b Fp(2)e Fs(g)2085 4412 y Fn(0)2121 4400 y Fs(U)1656
4500 y(P)35 b(;)50 b Ft(if)38 b Fs(g)26 b Fp(2)d Fs(H)7
b Fp(n)p Fs(U)1656 4600 y(S)28 b(;)50 b Ft(if)38 b Fs(g)26
b Fp(2)d Fs(G)p Fp(n)p Fu(\()p Fs(H)i Fp([)19 b Fs(g)2387
4612 y Fn(0)2424 4600 y Fs(U)9 b Fu(\))94 4775 y Ft(Pr)l(o)l(of)p
Fu(.)76 b(Let)40 b Fs(\036)595 4787 y Fo(O)r(; )757 4775
y Fu(b)r(e)g(a)g(co)r(cycle)e(of)i(t)n(yp)r(e)g Fs(I)7
b Fu(.)74 b(Then)40 b Fs(G=U)2153 4753 y Fp(\030)2153
4779 y Fu(=)2261 4775 y Fl(Z)p Fs(=)o Fu(2)20 b Fp(\010)27
b Fl(Z)o Fs(=)o Fu(2)34 b(and)40 b(im)n(\()p Fs( )s Fu(\))k(=)f
Fp(f)p Fs(O)r(;)14 b(P)e Fp(g)p Fu(.)73 b(If)94 4875
y Fs(g)26 b Fp(2)e Fs(G)p Fp(n)p Fu(\()p Fs(H)12 b Fp([)5
b Fs(g)559 4887 y Fn(0)596 4875 y Fs(U)k Fu(\),)23 b(then)e
Fs(g)965 4839 y Fm(\000)p Fn(1)962 4897 y(0)1054 4875
y Fs(g)26 b Fp(2)d Fs(H)7 b Fp(n)p Fs(U)i Fu(.)34 b(Using)20
b(form)n(ula)e(\(11\))i(one)h(can)g(compute)f Fs(\036)2803
4887 y Fo(O)r(; )2925 4875 y Fu(\()p Fs(g)s Fu(\))k(=)e
Fs( )s Fu(\()p Fs(g)3275 4839 y Fm(\000)p Fn(1)3272 4897
y(0)3364 4875 y Fs(g)s Fu(\))h(=)g Fs(P)12 b Fu(.)94
4974 y(Moreo)n(v)n(er,)25 b(for)i(an)n(y)g Fs(u)c Fp(2)g
Fs(U)37 b Fu(w)n(e)27 b(ha)n(v)n(e)f Fs(\036)1371 4986
y Fo(O)r(; )1494 4974 y Fu(\()p Fs(g)1566 4986 y Fn(0)1603
4974 y Fs(u)p Fu(\))d(=)g Fs( )s Fu(\()p Fs(g)1926 4939
y Fm(\000)p Fn(1)1923 4996 y(0)2015 4974 y Fs(g)2055
4986 y Fn(0)2091 4974 y Fs(u)p Fu(\))g(=)g Fs( )s Fu(\()p
Fs(u)p Fu(\))g(=)g Fs(O)r Fu(.)219 5074 y(Let)k Fs(\036)416
5086 y Fo(T)5 b(; )557 5074 y Fu(b)r(e)27 b(a)g(co)r(cycle)e(of)h(t)n
(yp)r(e)h Fs(I)7 b(I)g Fu(.)37 b(Fix)25 b(the)i(isomorphism)22
b Fs(G=U)2417 5052 y Fp(\030)2417 5078 y Fu(=)2505 5074
y Fl(Z)o Fs(=)p Fu(4)e(suc)n(h)26 b(that)p 3035 5028
77 4 v 27 w Fs(g)3075 5086 y Fn(0)3139 5074 y Fu(maps)f(in)n(to)g(the)
94 5174 y(1)6 b(+)g Fl(Z)n Fu(.)29 b(Sometimes)18 b(it)j(is)f(con)n(v)n
(enien)n(t)f(to)i(iden)n(tify)f(the)h(group)f Fs(G=U)30
b Fu(and)21 b Fl(Z)o Fs(=)p Fu(4)15 b(b)n(y)21 b(means)e(of)i(this)g
(isomorphism.)94 5273 y(An)n(y)28 b Fs(g)e Fp(2)d Fs(G)28
b Fu(can)f(b)r(e)h(written)f(in)g(the)h(form)1105 5448
y Fs(g)e Fu(=)c Fs(g)1301 5414 y Fo(i)1298 5468 y Fn(0)1335
5448 y Fs(u)37 b Fu(where)f Fs(i)23 b Fp(2)g(f)p Fu(0)p
Fs(;)14 b Fu(1)p Fs(;)g Fu(2)p Fs(;)g Fu(3)p Fp(g)34
b Fu(and)i Fs(u)23 b Fp(2)g Fs(U)32 b(;)1825 5697 y Fu(10)p
eop
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11 10 bop 94 98 a Fu(where)26 b Fs(i)g Fu(is)f(the)h
Ft(or)l(der)h Fu(of)g(an)e(elemen)n(t)g Fs(g)j Fu(under)e(the)h(elemen)
n(t)d Fs(g)2128 110 y Fn(0)2165 98 y Fu(.)36 b(If)27
b(the)g(order)d Fs(i)i Fu(of)g Fs(g)j Fu(is)c(ev)n(en,)h(then)h
Fs(g)3404 68 y Fo(i)3401 118 y Fn(0)3461 98 y Fp(2)c
Fs(H)7 b Fu(.)94 197 y(Using)27 b(form)n(ula)e(11)h(one)h(can)h(obtain)
e(that)1025 425 y Fs(\036)1074 437 y Fo(T)5 b(; )1189
425 y Fu(\()p Fs(g)s Fu(\))23 b(=)g Fs( )s Fu(\()p Fs(g)1539
391 y Fo(i)1536 446 y Fn(0)1573 425 y Fs(u)p Fu(\))g(=)f
Fs( )s Fu(\()p Fs(g)1895 391 y Fo(i)1892 446 y Fn(0)1930
425 y Fu(\))h(=)2073 308 y Fi(\032)2176 375 y Fs(O)j(;)51
b Fu(for)36 b Fs(i)23 b Fu(=)f(0)2176 474 y Fs(P)35 b(;)51
b Fu(for)36 b Fs(i)23 b Fu(=)g(2)94 664 y(If)29 b Fs(i)e
Fu(is)f(o)r(dd,)i(then)g Fs(g)736 634 y Fo(i)733 685
y Fn(0)793 664 y Fp(2)23 b Fs(G)p Fp(n)p Fs(H)35 b Fu(and)27
b(hence)h Fs(g)d Fu(=)e Fs(g)1670 634 y Fo(i)1667 685
y Fn(0)1704 664 y Fs(u)f Fp(2)i Fs(G)p Fp(n)p Fs(H)7
b Fu(.)36 b(Using)27 b(the)h(same)e(lemma)e(w)n(e)k(ha)n(v)n(e)1122
847 y Fs(\036)1171 859 y Fo(T)5 b(; )1286 847 y Fu(\()p
Fs(g)s Fu(\))23 b(=)f Fs( )s Fu(\()p Fs(g)1635 811 y
Fm(\000)p Fn(1)1632 869 y(0)1724 847 y Fs(g)s Fu(\))d(+)f
Fs(T)34 b Fu(=)22 b Fs( )s Fu(\()p Fs(g)2203 811 y Fm(\000)p
Fn(1)2200 869 y(0)2293 847 y Fs(g)2336 812 y Fo(i)2333
867 y Fn(0)2369 847 y Fs(u)p Fu(\))d(+)f Fs(T)1305 1029
y Fu(=)k Fs( )s Fu(\()p Fs(g)1524 994 y Fo(i)p Fm(\000)p
Fn(1)1521 1052 y(0)1637 1029 y Fs(u)p Fu(\))c(+)g Fs(T)34
b Fu(=)23 b Fs( )s Fu(\()p Fs(g)2121 994 y Fo(i)p Fm(\000)p
Fn(1)2118 1052 y(0)2234 1029 y Fu(\))18 b(+)g Fs(T)1508
1230 y Fu(=)1595 1113 y Fi(\032)1699 1179 y Fs(T)34 b(;)51
b Fu(for)36 b Fs(i)23 b Fu(=)g(1)1699 1279 y Fs(S)28
b(;)51 b Fu(for)36 b Fs(i)23 b Fu(=)f(3)p 3585 1367 57
4 v 3585 1417 4 50 v 3638 1417 V 3585 1420 57 4 v 219
1566 a(No)n(w)31 b(let)f(us)h(consider)e(co)r(cycles)g
Fs(\036)1343 1578 y Fo(O)r(; )1497 1566 y Fu(of)i(t)n(yp)r(e)g
Fs(I)7 b(I)g(I)g Fu(.)47 b(It)32 b(follo)n(ws)c(from)h(the)j
(de\014nition)d(that)i(ev)n(ery)f(suc)n(h)94 1666 y(co)r(cyle)h(is)f
(completely)f(determined)h(b)n(y)h(a)g(subgroup)g Fs(U)40
b Fu(of)32 b(index)f(2)g(in)g Fs(H)38 b Fu(suc)n(h)32
b(that)g Fs(U)38 b Fp(6)p Fu(=)30 b Fs(g)3166 1630 y
Fm(\000)p Fn(1)3163 1688 y(0)3254 1666 y Fs(U)9 b(g)3360
1678 y Fn(0)3397 1666 y Fu(.)49 b(Here)94 1766 y Fs(g)134
1778 y Fn(0)194 1766 y Fp(2)24 b Fs(G)p Fp(n)p Fs(H)k
Fu(and)22 b Fs(g)676 1735 y Fn(2)673 1786 y(0)736 1766
y Fp(2)h Fs(U)871 1778 y Fo(O)950 1766 y Fu(=)g Fs(U)15
b Fp(\\)7 b Fs(g)1215 1730 y Fm(\000)p Fn(1)1212 1788
y(0)1304 1766 y Fs(U)i(g)1410 1778 y Fn(0)1447 1766 y
Fu(.)35 b(If)22 b Fs( )1636 1778 y Fo(T)1710 1766 y Fu(and)g
Fs( )1920 1778 y Fo(S)1990 1766 y Fu(are)e(the)i(comp)r(ositions)c(of)k
Fs( )j Fu(with)c(the)h(pro)5 b(jections)94 1865 y(of)202
1877 y Fn(2)258 1844 y Fu(\026)239 1865 y Fs(E)32 b Fu(on)26
b Fp(f)p Fs(O)r(;)14 b(T)e Fp(g)25 b Fu(and)h Fp(f)p
Fs(O)r(;)14 b(S)5 b Fp(g)26 b Fu(resp)r(ectiv)n(ely)-7
b(,)24 b(then)j Fs( )f Fu(=)d Fs( )2024 1877 y Fo(T)2093
1865 y Fu(+)16 b Fs( )2228 1877 y Fo(S)2302 1865 y Fu(and)27
b Fs(U)32 b Fu(=)22 b(k)n(er)o(\()p Fs( )2836 1877 y
Fo(T)2889 1865 y Fu(\),)27 b Fs(g)3014 1830 y Fm(\000)p
Fn(1)3011 1887 y(0)3103 1865 y Fs(U)9 b(g)3209 1877 y
Fn(0)3268 1865 y Fu(=)23 b(k)n(er)o(\()p Fs( )3553 1877
y Fo(T)3606 1865 y Fu(\))94 1965 y(\(see)28 b(Section)f(3.3\).)36
b(Let)1535 2064 y Fs(U)1592 2076 y Fo(S)1663 2064 y Fu(=)23
b Fs(U)9 b Fp(n)p Fu(\()p Fs(g)1934 2029 y Fm(\000)p
Fn(1)1931 2087 y(0)2022 2064 y Fs(U)g(g)2128 2076 y Fn(0)2165
2064 y Fu(\))1533 2214 y Fs(U)1590 2226 y Fo(T)1665 2214
y Fu(=)23 b(\()p Fs(g)1828 2178 y Fm(\000)p Fn(1)1825
2236 y(0)1917 2214 y Fs(U)9 b(g)2023 2226 y Fn(0)2060
2214 y Fu(\))p Fp(n)p Fs(U)1448 2363 y(U)1505 2375 y
Fo(P)1583 2363 y Fu(=)22 b Fs(H)7 b Fp(n)p Fu(\()p Fs(U)27
b Fp([)19 b Fs(g)2021 2328 y Fm(\000)p Fn(1)2018 2385
y(0)2110 2363 y Fs(U)9 b(g)2216 2375 y Fn(0)2252 2363
y Fu(\))94 2513 y(b)r(e)28 b(the)f(cosets)f(in)g Fs(H)r(=U)855
2525 y Fo(O)910 2513 y Fu(.)37 b(Then)27 b Fs(G=U)1350
2525 y Fo(O)1429 2513 y Fu(=)c Fp(f)p Fs(U)1616 2525
y Fo(O)1671 2513 y Fs(;)14 b(U)1765 2525 y Fo(S)1813
2513 y Fs(;)g(U)1907 2525 y Fo(T)1959 2513 y Fs(;)g(U)2053
2525 y Fo(P)2108 2513 y Fs(;)g(g)2185 2525 y Fn(0)2221
2513 y Fs(U)2278 2525 y Fo(O)2335 2513 y Fs(;)g(g)2412
2525 y Fn(0)2448 2513 y Fs(U)2505 2525 y Fo(S)2553 2513
y Fs(;)g(g)2630 2525 y Fn(0)2667 2513 y Fs(U)2724 2525
y Fo(T)2776 2513 y Fs(;)g(g)2853 2525 y Fn(0)2890 2513
y Fs(U)2947 2525 y Fo(P)3002 2513 y Fs(;)g Fp(g)p Fu(.)36
b(Sc)n(hematically)94 2612 y(one)28 b(can)f(describ)r(e)f(the)i
(quotien)n(t)f(group)f Fs(G=U)1590 2624 y Fo(O)1674 2612
y Fu(b)n(y)h(the)h(diagram)p 1577 2822 579 4 v 1575 2922
4 100 v 1865 2922 V 2154 2922 V 1575 3021 V 1669 2992
a Fs(U)1726 3004 y Fo(S)p 1865 3021 V 1955 2992 a Fs(U)2012
3004 y Fo(P)p 2154 3021 V 1575 3121 V 1865 3121 V 2154
3121 V 1577 3124 579 4 v 1575 3224 4 100 v 1865 3224
V 2154 3224 V 1575 3324 V 1665 3294 a Fs(U)1722 3306
y Fo(O)p 1865 3324 V 1956 3294 a Fs(U)2013 3306 y Fo(T)p
2154 3324 V 1575 3423 V 1865 3423 V 2154 3423 V 1577
3427 579 4 v 1577 3443 V 1575 3543 4 100 v 1865 3543
V 2154 3543 V 1575 3642 V 1631 3613 a Fs(g)1671 3625
y Fn(0)1708 3613 y Fs(U)1765 3625 y Fo(S)p 1865 3642
V 1916 3613 a Fs(g)1956 3625 y Fn(0)1993 3613 y Fs(U)2050
3625 y Fo(P)p 2154 3642 V 1575 3742 V 1865 3742 V 2154
3742 V 1577 3745 579 4 v 1575 3845 4 100 v 1865 3845
V 2154 3845 V 1575 3945 V 1627 3915 a Fs(g)1667 3927
y Fn(0)1704 3915 y Fs(U)1761 3927 y Fo(O)p 1865 3945
V 1918 3915 a Fs(g)1958 3927 y Fn(0)1995 3915 y Fs(U)2052
3927 y Fo(T)p 2154 3945 V 1575 4044 V 1865 4044 V 2154
4044 V 1577 4048 579 4 v 94 4291 a Fu(Here)c(the)g(top)f(four)h
(squares)e(corresp)r(ond)f(to)j(the)g(group)e Fs(H)31
b Fu(and)23 b(all)f(eigh)n(t)g(squares)g(corresp)r(ond)g(to)h(the)h
(whole)94 4391 y(group)j Fs(G)p Fu(.)94 4573 y Fv(Prop)s(osition)k
(3.4.2)39 b Ft(L)l(et)30 b Fs(\036)23 b Fu(=)g Fs(\036)1201
4585 y Fo(O)r(; )1353 4573 y Ft(b)l(e)30 b(a)g(c)l(o)l(cycle)h(of)g
(typ)l(e)f Fs(I)7 b(I)g(I)g Ft(.)38 b(L)l(et)1538 4756
y Fs(V)1586 4768 y Fo(O)1666 4756 y Fu(=)22 b Fs(U)1810
4768 y Fo(O)1885 4756 y Fp([)d Fs(g)1999 4768 y Fn(0)2035
4756 y Fs(U)2092 4768 y Fo(O)2172 4756 y Fs(;)1548 4939
y(V)1596 4951 y Fo(S)1668 4939 y Fu(=)j Fs(U)1812 4951
y Fo(S)1878 4939 y Fp([)d Fs(g)1992 4951 y Fn(0)2029
4939 y Fs(U)2086 4951 y Fo(T)2161 4939 y Fs(;)1546 5088
y(V)1594 5100 y Fo(T)1670 5088 y Fu(=)j Fs(U)1814 5100
y Fo(T)1885 5088 y Fp([)d Fs(g)1999 5100 y Fn(0)2035
5088 y Fs(U)2092 5100 y Fo(S)2163 5088 y Fs(;)94 5238
y Ft(and)1539 5337 y Fs(V)1587 5349 y Fo(P)1666 5337
y Fu(=)k Fs(U)1811 5349 y Fo(P)1884 5337 y Fp([)c Fs(g)1998
5349 y Fn(0)2035 5337 y Fs(U)2092 5349 y Fo(P)2170 5337
y Fs(:)1825 5697 y Fu(11)p eop
%%Page: 12 12
12 11 bop 94 98 a Ft(Then)33 b Fs(V)361 110 y Fo(O)449
98 y Ft(is)f(a)g(\(not)g(normal\))g(sub)l(gr)l(oup)f(in)h
Fs(G)g Ft(of)g(index)g Fu(4)p Ft(.)44 b(The)33 b(sets)e
Fs(V)2479 110 y Fo(O)2536 98 y Ft(,)h Fs(V)2641 110 y
Fo(S)2690 98 y Ft(,)h Fs(V)2796 110 y Fo(T)2848 98 y
Ft(,)g Fs(V)2954 110 y Fo(P)3041 98 y Ft(ar)l(e)f(left)g(c)l(osets)g
(of)94 197 y(the)e(gr)l(oup)g Fs(G)g Ft(by)h Fs(V)712
209 y Fo(O)769 197 y Ft(.)38 b(The)31 b(c)l(o)l(cycle)g
Fs(\036)1329 209 y Fo(O)r(; )1481 197 y Ft(c)l(an)f(b)l(e)g(c)l(ompute)
l(d)g(by)g(the)g(formula)1338 533 y Fs(\036)1387 545
y Fo(O)r(; )1510 533 y Fu(\()p Fs(g)s Fu(\))23 b(=)1728
313 y Fi(8)1728 387 y(>)1728 412 y(>)1728 437 y(<)1728
587 y(>)1728 611 y(>)1728 636 y(:)1843 382 y Fs(O)j(;)51
b Ft(if)37 b Fs(g)26 b Fp(2)e Fs(V)2286 394 y Fo(O)1843
482 y Fs(S)k(;)51 b Ft(if)38 b Fs(g)25 b Fp(2)f Fs(V)2276
494 y Fo(S)1843 582 y Fs(T)34 b(;)51 b Ft(if)38 b Fs(g)26
b Fp(2)d Fs(V)2280 594 y Fo(T)1843 681 y Fs(P)35 b(;)51
b Ft(if)38 b Fs(g)25 b Fp(2)f Fs(V)2285 693 y Fo(P)94
865 y Ft(Pr)l(o)l(of)p Fu(.)53 b(W)-7 b(e)33 b(kno)n(w)e(that)i
Fs(G=U)1080 877 y Fo(O)1166 842 y Fp(\030)1166 869 y
Fu(=)1262 865 y Fs(D)1331 877 y Fn(4)1368 865 y Fu(.)51
b(Then)33 b(this)e(prop)r(osition)f(can)h(b)r(e)i(pro)n(v)n(ed)e(b)n(y)
h(direct)f(computation)94 964 y(with)d(using)e(the)i(form)n(ula)1147
1111 y Fs(\036)1196 1123 y Fo(O)r(; )1319 1111 y Fu(\()p
Fs(g)s Fu(\))23 b(=)1537 994 y Fi(\032)1640 1058 y Fs( )s
Fu(\()p Fs(g)s Fu(\))14 b Fs(;)320 b(g)25 b Fp(2)f Fs(H)1640
1161 y(g)1680 1173 y Fn(0)1717 1161 y Fs( )s Fu(\()p
Fs(g)1849 1126 y Fm(\000)p Fn(1)1846 1183 y(0)1938 1161
y Fs(g)s Fu(\))14 b Fs(;)111 b(g)25 b Fp(2)f Fs(G)p Fp(n)p
Fs(H)30 b(:)3490 1111 y Fu(\(13\))94 1312 y(and)f(rules)e(of)i(m)n
(ultiplication)23 b(in)28 b Fs(G=U)1349 1324 y Fo(O)1430
1289 y Fp(\030)1430 1316 y Fu(=)1520 1312 y Fs(D)1589
1324 y Fn(4)1626 1312 y Fu(.)40 b(F)-7 b(or)28 b(example,)f(let)h(us)h
(pro)n(v)n(e)e(that)i Fs(V)2872 1324 y Fo(O)2958 1312
y Fu(is)e(a)i(subgroup)e(in)i Fs(G)94 1411 y Fu(and)k(that)f(it)g(is)f
(not)h(normal)d(in)i Fs(G)p Fu(.)51 b(By)32 b(de\014nition)f(of)h(a)g
(co)r(cycle)e(of)j(t)n(yp)r(e)f Fs(I)7 b(I)g(I)g Fu(,)33
b(w)n(e)f(ha)n(v)n(e)f(that)p 3255 1366 77 4 v 33 w Fs(g)3295
1423 y Fn(0)3332 1379 y(2)3400 1411 y Fu(=)f(1)i(in)94
1511 y Fs(G=U)258 1523 y Fo(O)314 1511 y Fu(.)44 b(Hence)30
b Fs(V)678 1523 y Fo(O)762 1511 y Fu(=)c Fs(U)910 1523
y Fo(O)986 1511 y Fp([)20 b Fs(g)1101 1523 y Fn(0)1138
1511 y Fs(U)1195 1523 y Fo(O)1281 1511 y Fu(is)29 b(a)g(subgroup)g(in)g
Fs(G)p Fu(.)44 b(Let)30 b Fs(h)2231 1523 y Fo(S)2306
1511 y Fp(2)d Fs(U)2445 1523 y Fo(S)2493 1511 y Fu(,)j
Fs(h)2594 1523 y Fo(T)2673 1511 y Fp(2)d Fs(U)2812 1523
y Fo(T)2894 1511 y Fu(and)j Fs(h)3106 1523 y Fo(P)3188
1511 y Fp(2)d Fs(U)3327 1523 y Fo(P)3382 1511 y Fu(.)44
b(Then)94 1610 y(for)28 b(an)n(y)e Fs(u)d Fp(2)g Fs(U)584
1622 y Fo(O)668 1610 y Fu(w)n(e)k(ha)n(v)n(e:)p 649 1725
358 4 v 649 1807 a Fs(h)697 1772 y Fm(\000)p Fn(1)697
1832 y Fo(S)786 1807 y Fs(g)826 1819 y Fn(0)863 1807
y Fs(uh)959 1819 y Fo(S)1029 1807 y Fu(=)p 1117 1740
96 4 v 23 w Fs(h)1165 1819 y Fo(S)p 1213 1762 77 4 v
1213 1807 a Fs(g)1253 1819 y Fn(0)p 1290 1740 96 4 v
1290 1807 a Fs(h)1338 1819 y Fo(S)1409 1807 y Fu(=)p
1496 1762 77 4 v 22 w Fs(g)1536 1819 y Fn(0)1573 1807
y Fu(\()p 1605 1762 V Fs(g)1645 1819 y Fn(0)1682 1773
y Fm(\000)p Fn(1)p 1771 1740 96 4 v 1771 1807 a Fs(h)1819
1819 y Fo(S)p 1867 1762 77 4 v 1867 1807 a Fs(g)1907
1819 y Fn(0)1944 1807 y Fu(\))p 1976 1740 96 4 v Fs(h)2024
1819 y Fo(S)2095 1807 y Fu(=)p 2183 1762 77 4 v 23 w
Fs(g)2223 1819 y Fn(0)p 2260 1740 101 4 v 2260 1807 a
Fs(h)2308 1819 y Fo(T)p 2360 1740 96 4 v 2360 1807 a
Fs(h)2408 1819 y Fo(S)2479 1807 y Fu(=)p 2567 1762 77
4 v 23 w Fs(g)2607 1819 y Fn(0)p 2643 1740 104 4 v 2643
1807 a Fs(h)2691 1819 y Fo(P)2770 1807 y Fu(=)p 2857
1740 181 4 v 22 w Fs(g)2897 1819 y Fn(0)2934 1807 y Fs(h)2982
1819 y Fo(P)3060 1807 y Fs(:)94 1990 y Fu(In)j(other)e(w)n(ords,)g
Fs(h)729 1955 y Fm(\000)p Fn(1)729 2015 y Fo(S)818 1990
y Fs(g)858 2002 y Fn(0)895 1990 y Fs(uh)991 2002 y Fo(S)1064
1990 y Fp(2)e Fs(g)1185 2002 y Fn(0)1222 1990 y Fs(U)1279
2002 y Fo(P)1334 1990 y Fu(.)41 b(Hence)29 b Fs(g)1689
1955 y Fm(\000)p Fn(1)1686 2012 y(0)1778 1990 y Fs(V)1826
2002 y Fo(O)1883 1990 y Fs(g)1923 2002 y Fn(0)1985 1990
y Fp(6\032)c Fs(V)2123 2002 y Fo(O)2210 1990 y Fu(\(b)r(ecause)j(b)n(y)
h(de\014nition)f(of)h(the)g(set)g Fs(V)3457 2002 y Fo(O)3543
1990 y Fu(w)n(e)94 2090 y(ha)n(v)n(e)e(that)h Fs(V)514
2102 y Fo(O)589 2090 y Fp(\\)19 b Fs(g)703 2102 y Fn(0)740
2090 y Fs(U)797 2102 y Fo(P)875 2090 y Fu(=)j Fp(;)p
Fu(\).)p 3585 2040 57 4 v 3585 2090 4 50 v 3638 2090
V 3585 2093 57 4 v 219 2239 a(W)-7 b(e)28 b(can)f(illustrate)d(the)k(v)
-5 b(alues)27 b(of)g(co)r(cycle)f Fs(\036)1685 2251 y
Fo(O)r(; )1835 2239 y Fu(of)i(t)n(yp)r(e)g Fs(I)7 b(I)g(I)35
b Fu(b)n(y)27 b(the)h(diagram)p 1530 2358 672 4 v 1528
2457 4 100 v 1864 2457 V 2200 2457 V 1528 2557 V 1580
2527 a Fs(S)p 1864 2557 V 285 w(P)p 2200 2557 V 1528
2656 V 1864 2656 V 2200 2656 V 1530 2660 672 4 v 1528
2759 4 100 v 1864 2759 V 2200 2759 V 1528 2859 V 1580
2829 a(O)p 1864 2859 V 273 w(T)p 2200 2859 V 1528 2959
V 1864 2959 V 2200 2959 V 1530 2962 672 4 v 1530 2979
V 1528 3078 4 100 v 1864 3078 V 2200 3078 V 1528 3178
V 1580 3148 a(T)p 1864 3178 V 287 w(P)p 2200 3178 V 1528
3277 V 1864 3277 V 2200 3277 V 1530 3281 672 4 v 1528
3380 4 100 v 1864 3380 V 2200 3380 V 1528 3480 V 1580
3450 a(O)p 1864 3480 V 273 w(S)p 2200 3480 V 1528 3580
V 1864 3580 V 2200 3580 V 1530 3583 672 4 v 94 3954 a
Fk(3.5)113 b Fa(\016)39 b Fj(on)34 b(c)-5 b(o)g(cycles)94
4107 y Fu(No)n(w)26 b(w)n(e)f(w)n(an)n(t)g(to)h(lift)e(the)i
(cohomology)21 b(classes)j(of)h(co)r(cycles)f Fs(\036)2146
4119 y Fo(Q; )2295 4107 y Fu(to)h(the)h(group)2769 4119
y Fn(2)2806 4107 y Fu(Br)o(\()p Fs(E)5 b Fu(\).)37 b(In)26
b(other)f(w)n(ords,)94 4207 y(w)n(e)g(w)n(an)n(t)e(to)i(describ)r(e)e
(unrami\014ed)f(cen)n(tral)h(simple)e(algebras)h(of)i(an)g(exp)r(onen)n
(t)h(2)f(suc)n(h)g(that)g(its)g(classes)e(maps)94 4307
y(in)n(to)27 b(classes)e(cls)o(\()p Fs(\036)701 4319
y Fo(Q; )824 4307 y Fu(\))e Fp(2)h Fs(H)1034 4277 y Fn(1)1071
4307 y Fu(\()p Fs(G;)1224 4286 y Fu(\026)1205 4307 y
Fs(E)5 b Fu(\))28 b(under)f(the)h(homomorphism)23 b Fs(\031)s
Fu(.)219 4406 y(According)16 b(to)j(Section)e(2.3)h(and)g(form)n(ula)d
(6)j(the)h(\014rst)f(step)g(m)n(ust)g(b)r(e)h(computing)d(the)j(elemen)
n(ts)d Fs(\016)s Fu(\(cls)o(\()p Fs(\036)3484 4418 y
Fo(Q; )3607 4406 y Fu(\)\))94 4506 y(for)29 b(cycles)e(of)i(the)g
Fs(I)36 b Fu(and)29 b Fs(I)7 b(I)36 b Fu(t)n(yp)r(es.)k(W)-7
b(e)30 b(will)c(determine)h Fs(\016)s Fu(\(cls)o(\()p
Fs(\036)2245 4518 y Fo(Q; )2368 4506 y Fu(\)\))i(in)f(terms)g(of)h
(explicit)d(2-co)r(cycles)g(of)94 4606 y(the)i(group)f
Fs(G)h Fu(with)f(co)r(e\016cien)n(ts)f(in)1272 4614 y(P)1329
4606 y(\()1380 4585 y(\026)1361 4606 y Fs(E)5 b Fu(\).)219
4705 y(Consider)26 b(the)i(follo)n(wing)23 b(particular)h(case)j(of)h
(the)g(short)f(exact)g(sequence)g(\(5\):)1216 4898 y(0)c
Fp(!)1387 4906 y Fu(P)1443 4898 y(\()1495 4877 y(\026)1475
4898 y Fs(E)6 b Fu(\))23 b Fp(\000)-14 b(!)23 b Fu(Div)1885
4861 y Fn(0)1923 4898 y Fu(\()1974 4877 y(\026)1955 4898
y Fs(E)5 b Fu(\))2084 4851 y Fn(sum)2076 4898 y Fp(\000)-14
b(!)2252 4877 y Fu(\026)2233 4898 y Fs(E)28 b Fp(!)23
b Fu(0)g Fs(:)94 5080 y Fu(F)-7 b(or)27 b(an)n(y)g(p)r(oin)n(t)g
Fs(W)35 b Fp(2)828 5059 y Fu(\026)808 5080 y Fs(E)e Fu(denote)28
b(b)n(y)f(\()p Fs(W)12 b Fu(\))28 b(the)g(same)e(p)r(oin)n(t)h(view)n
(ed)f(as)h(a)g(divisor)e(on)i Fs(E)5 b Fu(.)37 b(Then)1448
5263 y(sum)o(\(\(W\))20 b Fp(\000)e Fu(\(O\)\))23 b(=)g(W)h
Fs(:)1825 5697 y Fu(12)p eop
%%Page: 13 13
13 12 bop 94 98 a Fu(Let)36 b Fs(\036)g Fu(b)r(e)g(a)f(con)n(tin)n
(uous)f(\(in)h(sense)g(of)h(the)f(pro\014nite)g(top)r(ology)e(on)i
Fs(G)p Fu(\))h(1-co)r(cycle)e(of)h(the)h(group)e Fs(G)i
Fu(with)94 206 y(co)r(e\016cien)n(ts)27 b(in)631 185
y(\026)612 206 y Fs(E)5 b Fu(.)37 b(Denote)27 b(b)n(y)1149
185 y(^)1138 206 y Fs(\036)h Fu(the)g(con)n(tin)n(uous)d(map)1545
341 y(^)1535 363 y Fs(\036)e Fu(:)g Fs(G)h Fp(\000)-15
b(!)23 b Fu(Div)2030 327 y Fn(0)2067 363 y Fu(\()2119
342 y(\026)2099 363 y Fs(E)5 b Fu(\))94 520 y(suc)n(h)28
b(that)1517 598 y(^)1506 620 y Fs(\036)q Fu(\()p Fs(g)s
Fu(\))23 b(=)f(\()p Fs(\036)p Fu(\()p Fs(g)s Fu(\)\))e
Fp(\000)e Fu(\()p Fs(O)r Fu(\))94 763 y(for)28 b(an)n(y)e
Fs(g)g Fp(2)d Fs(G)p Fu(.)38 b(Then)27 b Fs(@)924 741
y Fu(^)913 763 y Fs(\036)h Fu(is)f(a)g(2-co)r(cycle)e(of)j
Fs(G)g Fu(with)f(co)r(e\016cien)n(ts)f(in)2392 771 y(P)2448
763 y(\()2500 742 y(\026)2480 763 y Fs(E)6 b Fu(\),)28
b(and)1504 929 y Fs(\016)s Fu(\(cls)o(\()p Fs(\036)p
Fu(\)\))d(=)d(cls)o(\()p Fs(@)2111 907 y Fu(^)2100 929
y Fs(\036)q Fu(\))h Fs(:)94 1086 y Fu(The)28 b(form)n(ula)1169
1185 y Fs(@)1229 1163 y Fu(^)1218 1185 y Fs(\036)q Fu(\()p
Fs(g)1340 1197 y Fn(1)1377 1185 y Fs(;)14 b(g)1454 1197
y Fn(2)1491 1185 y Fu(\))23 b(=)g Fs(g)1674 1197 y Fn(1)1721
1163 y Fu(^)1710 1185 y Fs(\036)q Fu(\()p Fs(g)1832 1197
y Fn(2)1869 1185 y Fu(\))c Fp(\000)2014 1163 y Fu(^)2003
1185 y Fs(\036)p Fu(\()p Fs(g)2124 1197 y Fn(1)2161 1185
y Fs(g)2201 1197 y Fn(2)2238 1185 y Fu(\))g(+)2383 1163
y(^)2372 1185 y Fs(\036)p Fu(\()p Fs(g)2493 1197 y Fn(2)2531
1185 y Fu(\))927 b(\(14\))94 1328 y(allo)n(ws)32 b(to)j(compute)f
Fs(@)861 1306 y Fu(^)851 1328 y Fs(\036)h Fu(b)n(y)1068
1306 y(^)1058 1328 y Fs(\036)p Fu(.)59 b(W)-7 b(e)35
b(need)g(to)g(do)g(this)f(for)g(co)r(cycles)f(of)i(t)n(yp)r(es)g
Fs(I)7 b Fu(,)37 b Fs(I)7 b(I)42 b Fu(and)35 b Fs(I)7
b(I)g(I)g Fu(.)58 b(First)34 b(w)n(e)94 1428 y(consider)26
b(the)i(cases)f(of)g(co)r(cycles)f(of)h Fs(I)35 b Fu(and)27
b Fs(I)7 b(I)35 b Fu(t)n(yp)r(es.)94 1698 y Fw(4)135
b(Quaternion)45 b(and)f(cyclic)f(algebras)94 1897 y Fk(4.1)113
b Fj(A)n(lgebr)-5 b(as)34 b(c)-5 b(orr)g(esp)g(onding)33
b(to)i(c)-5 b(o)g(cyles)34 b(of)h(typ)-5 b(e)35 b Fa(I)94
2050 y Fv(Prop)s(osition)c(4.1.1)39 b Ft(L)l(et)32 b
Fs(\036)1043 2062 y Fo(O)r(; )1197 2050 y Ft(b)l(e)g(a)g(c)l(o)l(cycle)
h(of)g(typ)l(e)f Fs(I)7 b Ft(,)33 b Fs(U)i Fu(=)27 b(k)n(er)o(\()p
Fs( )s Fu(\))32 b Ft(and)g Fs(W)39 b Fu(=)27 b Fs(U)h
Fp([)21 b Fs(g)3053 2062 y Fn(0)3089 2050 y Fs(U)41 b
Ft(the)32 b(sub)l(gr)l(oup)94 2149 y(of)h(index)e Fu(2)g
Ft(in)h Fs(G)g Ft(\(se)l(e)f(Pr)l(op)l(osition)i(3.4.1\).)46
b(Consider)33 b(an)f(element)f Fs(a)26 b Fp(2)g Fs(k)s
Fp(n)p Fs(k)2655 2119 y Fn(2)2723 2149 y Ft(such)32 b(that)f
Fs(k)s Fu(\()3163 2090 y Fp(p)p 3233 2090 44 4 v 3233
2149 a Fs(a)o Fu(\))c(=)3428 2127 y(\026)3426 2149 y
Fs(k)3472 2119 y Fo(W)3578 2149 y Ft(is)94 2249 y(the)j(sub\014eld)g
(invariant)h(under)e(the)h(action)h(of)f Fs(W)12 b Ft(.)39
b(Then)30 b(the)g(quaternion)g(algebr)l(a)1668 2406 y
Fu(\()p Fs(a;)14 b(x)19 b Fp(\000)f Fs(x)1977 2418 y
Fo(P)2032 2406 y Fu(\))94 2563 y Ft(with)34 b(c)l(enter)e
Fs(k)s Fu(\()p Fs(E)5 b Fu(\))33 b Ft(is)g(unr)l(ami\014e)l(d.)46
b(The)34 b(class)f(of)g Fu(\()p Fs(a;)14 b(x)21 b Fp(\000)g
Fs(x)2072 2575 y Fo(P)2127 2563 y Fu(\))33 b Ft(maps)g(into)g(the)g
(class)g Fu(cls)o(\()p Fs(\036)3103 2575 y Fo(O)r(; )3225
2563 y Fu(\))g Ft(under)g(the)94 2662 y(homomorphism)g
Fs(\031)26 b Fu(:)d(Br\()p Fs(E)5 b Fu(\))23 b Fp(!)g
Fs(H)1223 2632 y Fn(1)1260 2662 y Fu(\()p Fs(G;)1414
2641 y Fu(\026)1394 2662 y Fs(E)6 b Fu(\))p Ft(.)94 2819
y(Pr)l(o)l(of)p Fu(.)39 b(Consider)25 b(the)j(2-co)r(cycle)d(\010)1252
2831 y Fo(P)r(;a)1379 2819 y Fu(:)e Fs(G)c Fp(\002)f
Fs(G)24 b Fp(!)1789 2797 y Fu(\026)1787 2819 y Fs(k)r
Fu(\()1884 2798 y(\026)1864 2819 y Fs(E)6 b Fu(\))1963
2789 y Fm(\003)2029 2819 y Fu(de\014ned)28 b(b)n(y)f(the)h(form)n(ula)
816 3029 y(\010)876 3041 y Fo(P)r(;a)980 3029 y Fu(\()p
Fs(g)1052 3041 y Fn(1)1089 3029 y Fs(;)14 b(g)1166 3041
y Fn(2)1203 3029 y Fu(\))23 b(=)1346 2912 y Fi(\032)1450
2978 y Fs(x)c Fp(\000)f Fs(x)1646 2990 y Fo(P)1725 2978
y Fs(;)50 b Fu(if)37 b(\()p Fs(g)1956 2990 y Fn(1)1993
2978 y Fs(;)14 b(g)2070 2990 y Fn(2)2107 2978 y Fu(\))23
b Fp(2)g Fu(\()p Fs(G)p Fp(n)p Fs(W)12 b Fu(\))19 b Fp(\002)f
Fu(\()p Fs(G)p Fp(n)p Fs(W)12 b Fu(\))1450 3078 y(1)22
b Fs(;)51 b Fu(in)27 b(the)h(other)f(case)94 3243 y(The)35
b(co)r(cycle)d(\010)624 3255 y Fo(P)r(;a)762 3243 y Fu(de\014nes)i(the)
g(quaternion)e(algebra)g(\()p Fs(a;)14 b(x)23 b Fp(\000)f
Fs(x)2229 3255 y Fo(P)2285 3243 y Fu(\))34 b(o)n(v)n(er)f
Fs(k)s Fu(\()p Fs(E)5 b Fu(\).)56 b(If)35 b(div)f(:)3088
3221 y(\026)3085 3243 y Fs(k)s Fu(\()3183 3222 y(\026)3163
3243 y Fs(E)6 b Fu(\))3262 3212 y Fm(\003)3334 3243 y
Fp(!)3451 3251 y Fu(P)3507 3243 y(\()3559 3222 y(\026)3539
3243 y Fs(E)g Fu(\))94 3342 y(is)32 b(the)h(map)e(sending)h
Fs(f)39 b Fp(2)995 3320 y Fu(\026)993 3342 y Fs(k)s Fu(\()1090
3321 y(\026)1071 3342 y Fs(E)5 b Fu(\))1169 3312 y Fm(\003)1240
3342 y Fu(in)n(to)32 b(the)h(divisor)c(div\()p Fs(f)9
b Fu(\))33 b(of)f(zeros)g(and)g(p)r(oles)f(of)i(the)g(function)f
Fs(f)9 b Fu(,)33 b(then)94 3451 y Fs(@)154 3429 y Fu(^)143
3451 y Fs(\036)192 3463 y Fo(O)r(; )338 3451 y Fu(=)23
b(div)14 b Fp(\016)p Fu(\010)656 3463 y Fo(P)r(;a)759
3451 y Fu(.)37 b(T)-7 b(aking)26 b(in)n(to)g(accoun)n(t)h(the)h
(diagram)c(in)j(Section)f(2.3,)h(one)g(can)g(see)h(that)211
3620 y(in)n(v)o(\(cls)o(\(\010)539 3632 y Fo(P)r(;a)644
3620 y Fu(\)\))23 b(=)g Fs(\014)t Fu(\()p Fs(\013)p Fu(\(cls\(\010)1173
3632 y Fo(P)r(;a)1277 3620 y Fu(\)\)\))h(=)f Fs(\014)t
Fu(\(cls)o(\(div)14 b Fp(\016)p Fu(\010)1923 3632 y Fo(P)r(;a)2027
3620 y Fu(\)\))23 b(=)g Fs(\014)t Fu(\(cls)o(\()p Fs(@)2470
3598 y Fu(^)2459 3620 y Fs(\036)2508 3632 y Fo(O)r(; )2631
3620 y Fu(\)\))h(=)e Fs(\014)t Fu(\()p Fs(\016)s Fu(\(cls\()p
Fs(\036)3136 3632 y Fo(O)r(; )3259 3620 y Fu(\)\))h(=)g(0)g
Fs(:)94 3777 y Fu(Hence)28 b(the)g(algebra)d(\()p Fs(a;)14
b(x)19 b Fp(\000)f Fs(x)1084 3789 y Fo(P)1140 3777 y
Fu(\))28 b(is)e(unrami\014ed)g(and)h Fs(\031)s Fu(\(cls)o(\(\010)2127
3789 y Fo(P)r(;a)2232 3777 y Fu(\)\))c(=)g(cls)o(\()p
Fs(\036)2581 3789 y Fo(O)r(; )2703 3777 y Fu(\).)p 3585
3727 57 4 v 3585 3777 4 50 v 3638 3777 V 3585 3780 57
4 v 94 4044 a Fk(4.2)113 b Fa(\016)39 b Fj(for)34 b(c)-5
b(o)g(cycles)34 b(of)h(typ)-5 b(e)35 b Fa(I)8 b(I)94
4197 y Fv(Prop)s(osition)31 b(4.2.1)39 b Ft(L)l(et)32
b Fs(\036)1043 4209 y Fo(T)5 b(; )1190 4197 y Ft(b)l(e)32
b(a)g(c)l(o)l(cycle)h(of)g(typ)l(e)f Fs(I)7 b(I)g Ft(.)45
b(Then)32 b(the)g Fu(2)p Ft(-c)l(o)l(cycle)g Fs(@)2849
4175 y Fu(^)2838 4197 y Fs(\036)2887 4209 y Fo(T)5 b(; )3029
4197 y Fu(:)27 b Fs(G)20 b Fp(\002)g Fs(G)27 b Fp(!)3451
4205 y Fu(P)3507 4197 y(\()3559 4176 y(\026)3539 4197
y Fs(E)6 b Fu(\))94 4297 y Ft(is)31 b(given)f(in)g(the)g(fol)t(lowing)i
(table)p 265 4477 3178 4 v 263 4543 4 67 v 898 4543 V
1196 4543 V 1944 4543 V 2693 4543 V 3441 4543 V 263 4609
V 315 4589 a Fo(g)347 4597 y Fh(2)398 4589 y Fm(2)19
b Fo(G)p Fm(n)p Fn(\()p Fo(H)h Fm([)14 b Fo(g)739 4597
y Fh(0)772 4589 y Fo(U)6 b Fn(\))p 898 4609 V 183 w(0)p
1196 4609 V 183 w(\()p Fo(S)s Fn(\))15 b Fm(\000)g Fn(\()p
Fo(T)9 b Fn(\))15 b(+)g(\()p Fo(P)9 b Fn(\))15 b Fm(\000)g
Fn(\()p Fo(O)r Fn(\))p 1944 4609 V 248 w(2\()p Fo(T)9
b Fn(\))15 b Fm(\000)g Fn(2\()p Fo(O)r Fn(\))p 2693 4609
V 248 w(\()p Fo(T)9 b Fn(\))15 b Fm(\000)g Fn(\()p Fo(P)9
b Fn(\))16 b(+)f(\()p Fo(S)s Fn(\))g Fm(\000)g Fn(\()p
Fo(O)r Fn(\))p 3441 4609 V 263 4676 V 898 4676 V 1196
4676 V 1944 4676 V 2693 4676 V 3441 4676 V 265 4679 3178
4 v 263 4746 4 67 v 898 4746 V 1196 4746 V 1944 4746
V 2693 4746 V 3441 4746 V 263 4812 V 450 4792 a Fo(g)482
4800 y Fh(2)534 4792 y Fm(2)k Fo(g)630 4800 y Fh(0)663
4792 y Fo(U)p 898 4812 V 324 w Fn(0)p 1196 4812 V 183
w(\()p Fo(T)9 b Fn(\))15 b Fm(\000)g Fn(\()p Fo(S)s Fn(\))g(+)g(\()p
Fo(P)9 b Fn(\))15 b Fm(\000)g Fn(\()p Fo(O)r Fn(\))p
1944 4812 V 100 w(\()p Fo(S)s Fn(\))g Fm(\000)g Fn(\()p
Fo(P)9 b Fn(\))15 b(+)g(\()p Fo(T)9 b Fn(\))15 b Fm(\000)g
Fn(\()p Fo(O)r Fn(\))p 2693 4812 V 250 w(2\()p Fo(S)s
Fn(\))g Fm(\000)g Fn(2\()p Fo(O)r Fn(\))p 3441 4812 V
263 4878 V 898 4878 V 1196 4878 V 1944 4878 V 2693 4878
V 3441 4878 V 265 4882 3178 4 v 263 4948 4 67 v 898 4948
V 1196 4948 V 1944 4948 V 2693 4948 V 3441 4948 V 263
5015 V 436 4995 a Fo(g)468 5003 y Fh(2)520 4995 y Fm(2)k
Fo(H)t Fm(n)p Fo(U)p 898 5015 V 311 w Fn(0)p 1196 5015
V 329 w(2\()p Fo(P)9 b Fn(\))16 b Fm(\000)f Fn(2\()p
Fo(O)r Fn(\))p 1944 5015 V 246 w(\()p Fo(P)9 b Fn(\))15
b Fm(\000)g Fn(\()p Fo(S)s Fn(\))g(+)g(\()p Fo(T)9 b
Fn(\))15 b Fm(\000)g Fn(\()p Fo(O)r Fn(\))p 2693 5015
V 100 w(\()p Fo(P)9 b Fn(\))15 b Fm(\000)g Fn(\()p Fo(T)9
b Fn(\))16 b(+)f(\()p Fo(S)s Fn(\))g Fm(\000)g Fn(\()p
Fo(O)r Fn(\))p 3441 5015 V 263 5081 V 898 5081 V 1196
5081 V 1944 5081 V 2693 5081 V 3441 5081 V 265 5084 3178
4 v 263 5151 4 67 v 898 5151 V 1196 5151 V 1944 5151
V 2693 5151 V 3441 5151 V 263 5217 V 483 5197 a Fo(g)515
5205 y Fh(2)566 5197 y Fm(2)k Fo(U)p 898 5217 V 357 w
Fn(0)p 1196 5217 V 490 w(0)p 1944 5217 V 716 w(0)p 2693
5217 V 715 w(0)p 3441 5217 V 263 5284 V 898 5284 V 1196
5284 V 1944 5284 V 2693 5284 V 3441 5284 V 265 5287 3178
4 v 263 5353 4 67 v 898 5353 V 1196 5353 V 1944 5353
V 2693 5353 V 3441 5353 V 263 5420 V 898 5420 V 949 5400
a Fo(g)981 5408 y Fh(1)1033 5400 y Fm(2)g Fo(U)p 1196
5420 V 284 w(g)1458 5408 y Fh(1)1510 5400 y Fm(2)g Fo(H)t
Fm(n)p Fo(U)p 1944 5420 V 477 w(g)2220 5408 y Fh(1)2272
5400 y Fm(2)g Fo(g)2368 5408 y Fh(0)2401 5400 y Fo(U)p
2693 5420 V 355 w(g)2833 5408 y Fh(1)2885 5400 y Fm(2)f
Fo(G)p Fm(n)p Fn(\()p Fo(H)i Fm([)15 b Fo(g)3226 5408
y Fh(0)3259 5400 y Fo(U)6 b Fn(\))p 3441 5420 V 263 5486
V 898 5486 V 1196 5486 V 1944 5486 V 2693 5486 V 3441
5486 V 265 5489 3178 4 v 1825 5697 a Fu(13)p eop
%%Page: 14 14
14 13 bop 94 98 a Ft(Pr)l(o)l(of)p Fu(.)39 b(By)27 b(Prop)r(osition)d
(3.4.1)i(w)n(e)h(ha)n(v)n(e)g(the)h(follo)n(wing)23 b(explicit)i(form)n
(ula)g(for)2666 76 y(^)2655 98 y Fs(\036)2704 110 y Fo(T)5
b(; )2819 98 y Fu(:)892 404 y(^)881 426 y Fs(\036)930
438 y Fo(T)g(; )1045 426 y Fu(\()p Fs(g)s Fu(\))23 b(=)1263
206 y Fi(8)1263 280 y(>)1263 305 y(>)1263 330 y(<)1263
480 y(>)1263 505 y(>)1263 530 y(:)1378 276 y Fu(0)g Fs(;)87
b(g)26 b Fp(2)d Fs(U)32 b Fp(,)23 b Fs(i)g Fu(=)g(0)1378
375 y(\()p Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r
Fu(\))24 b Fs(;)88 b(g)26 b Fp(2)d Fs(g)2052 387 y Fn(0)2089
375 y Fs(U)32 b Fp(,)23 b Fs(i)g Fu(=)f(1)1378 475 y(\()p
Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))24
b Fs(;)88 b(g)26 b Fp(2)d Fs(H)7 b Fp(n)p Fs(U)31 b Fp(,)23
b Fs(i)g Fu(=)f(2)1378 574 y(\()p Fs(S)5 b Fu(\))19 b
Fp(\000)f Fu(\()p Fs(O)r Fu(\))24 b Fs(;)88 b(g)25 b
Fp(2)f Fs(G)p Fp(n)p Fu(\()p Fs(H)h Fp([)19 b Fs(g)2355
586 y Fn(0)2392 574 y Fs(U)9 b Fu(\))23 b Fp(,)g Fs(i)g
Fu(=)f(3)3490 426 y(\(15\))219 710 y(Let)k Fs(g)406 722
y Fn(1)443 710 y Fs(;)14 b(g)520 722 y Fn(2)579 710 y
Fp(2)24 b Fs(G)i Fu(and)f Fs(i)937 722 y Fn(1)974 710
y Fu(,)h Fs(i)1052 722 y Fn(2)1114 710 y Fu(are)f(the)h(orders)e(of)h
Fs(g)1772 722 y Fn(1)1809 710 y Fu(,)h Fs(g)1898 722
y Fn(2)1960 710 y Fu(resp)r(ectiv)n(ely)-7 b(.)34 b(It)26
b(is)e(clear)g(that)h(the)h(order)e Fs(i)i Fu(of)f Fs(g)3524
722 y Fn(1)3561 710 y Fs(g)3601 722 y Fn(2)94 810 y Fu(is)32
b(equal)e(to)i Fs(i)541 822 y Fn(1)600 810 y Fu(+)21
b Fs(i)715 822 y Fn(2)784 810 y Fu(mo)r(dulo)30 b(4.)50
b(The)32 b(direct)g(computation)d(with)j(the)h(form)n(ulas)c(\(15\))j
(and)g(\(14\))f(giv)n(es)f(the)94 919 y(follo)n(wing)24
b(explicit)h(form)h(of)i(the)g(co)r(cycle)e Fs(@)1517
897 y Fu(^)1507 919 y Fs(\036)1556 931 y Fo(T)5 b(; )1670
919 y Fu(:)p 502 1115 2729 4 v 500 1215 4 100 v 552 1185
a Fs(i)581 1197 y Fn(1)p 666 1215 V 717 1185 a Fs(i)746
1197 y Fn(2)p 831 1215 V 883 1185 a Fs(g)923 1197 y Fn(1)960
1185 y Fs(\036)1009 1197 y Fo(T)g(; )1123 1185 y Fu(\()p
Fs(g)1195 1197 y Fn(2)1233 1185 y Fu(\))p 1313 1215 V
100 w Fp(\000)p Fs(\036)1479 1197 y Fo(T)g(; )1593 1185
y Fu(\()p Fs(g)1665 1197 y Fn(1)1702 1185 y Fs(g)1742
1197 y Fn(2)1779 1185 y Fu(\))p 1859 1215 V 128 w Fs(\036)1988
1197 y Fo(T)g(; )2103 1185 y Fu(\()p Fs(g)2175 1197 y
Fn(1)2212 1185 y Fu(\))p 2320 1215 V 458 w(sum)p 3229
1215 V 502 1218 2729 4 v 502 1235 V 500 1334 4 100 v
564 1305 a(0)p 666 1334 V 123 w(0)p 831 1334 V 282 w(0)p
1313 1334 V 472 w(0)p 1859 1334 V 462 w(0)p 2320 1334
V 642 w(0)p 3229 1334 V 502 1338 2729 4 v 500 1437 4
100 v 564 1408 a(0)p 666 1437 V 123 w(1)p 831 1437 V
125 w(\()p Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r
Fu(\))p 1313 1437 V 126 w Fp(\000)p Fu(\()p Fs(T)12 b
Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p 1859 1437 V 274 w(0)p
2320 1437 V 642 w(0)p 3229 1437 V 502 1441 2729 4 v 500
1540 4 100 v 564 1510 a(0)p 666 1540 V 123 w(2)p 831
1540 V 122 w(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p
Fs(O)r Fu(\))p 1313 1540 V 122 w Fp(\000)p Fu(\()p Fs(P)12
b Fu(\))19 b(+)f(\()p Fs(O)r Fu(\))p 1859 1540 V 271
w(0)p 2320 1540 V 642 w(0)p 3229 1540 V 502 1544 2729
4 v 500 1643 4 100 v 564 1613 a(0)p 666 1643 V 123 w(3)p
831 1643 V 127 w(\()p Fs(S)5 b Fu(\))19 b Fp(\000)f Fu(\()p
Fs(O)r Fu(\))p 1313 1643 V 131 w Fp(\000)p Fu(\()p Fs(S)5
b Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p 1859 1643 V 276
w(0)p 2320 1643 V 642 w(0)p 3229 1643 V 502 1647 2729
4 v 502 1663 V 500 1763 4 100 v 564 1733 a(1)p 666 1763
V 123 w(0)p 831 1763 V 282 w(0)p 1313 1763 V 282 w Fp(\000)p
Fu(\()p Fs(T)12 b Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p
1859 1763 V 116 w(\()p Fs(T)12 b Fu(\))18 b Fp(\000)g
Fu(\()p Fs(O)r Fu(\))p 2320 1763 V 487 w(0)p 3229 1763
V 502 1766 2729 4 v 500 1866 4 100 v 564 1836 a(1)p 666
1866 V 123 w(1)p 831 1866 V 127 w(\()p Fs(S)5 b Fu(\))19
b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 1313 1866 V 126 w
Fp(\000)p Fu(\()p Fs(P)12 b Fu(\))19 b(+)f(\()p Fs(O)r
Fu(\))p 1859 1866 V 113 w(\()p Fs(T)12 b Fu(\))18 b Fp(\000)g
Fu(\()p Fs(O)r Fu(\))p 2320 1866 V 104 w(\()p Fs(S)5
b Fu(\))18 b Fp(\000)g Fu(\()p Fs(P)12 b Fu(\))19 b(+)f(\()p
Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p
3229 1866 V 502 1869 2729 4 v 500 1969 4 100 v 564 1939
a(1)p 666 1969 V 123 w(2)p 831 1969 V 122 w(\()p Fs(P)12
b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 1313 1969
V 127 w Fp(\000)p Fu(\()p Fs(S)5 b Fu(\))18 b(+)g(\()p
Fs(O)r Fu(\))p 1859 1969 V 118 w(\()p Fs(T)12 b Fu(\))18
b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 2320 1969 V 104 w(\()p
Fs(P)12 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(S)5 b Fu(\))19
b(+)f(\()p Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r
Fu(\))p 3229 1969 V 502 1972 2729 4 v 500 2072 4 100
v 564 2042 a(1)p 666 2072 V 123 w(3)p 831 2072 V 125
w(\()p Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r
Fu(\))p 1313 2072 V 316 w(0)p 1859 2072 V 304 w(\()p
Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p
2320 2072 V 288 w(2\()p Fs(T)12 b Fu(\))18 b Fp(\000)g
Fu(2\()p Fs(O)r Fu(\))p 3229 2072 V 502 2075 2729 4 v
502 2092 V 500 2191 4 100 v 564 2161 a(2)p 666 2191 V
123 w(0)p 831 2191 V 282 w(0)p 1313 2191 V 280 w Fp(\000)p
Fu(\()p Fs(P)12 b Fu(\))19 b(+)f(\()p Fs(O)r Fu(\))p
1859 2191 V 111 w(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f
Fu(\()p Fs(O)r Fu(\))p 2320 2191 V 484 w(0)p 3229 2191
V 502 2195 2729 4 v 500 2294 4 100 v 564 2264 a(2)p 666
2294 V 123 w(1)p 831 2294 V 125 w(\()p Fs(T)12 b Fu(\))18
b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 1313 2294 V 129 w
Fp(\000)p Fu(\()p Fs(S)5 b Fu(\))18 b(+)g(\()p Fs(O)r
Fu(\))p 1859 2294 V 116 w(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f
Fu(\()p Fs(O)r Fu(\))p 2320 2294 V 101 w(\()p Fs(T)12
b Fu(\))18 b Fp(\000)g Fu(\()p Fs(S)5 b Fu(\))18 b(+)g(\()p
Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p
3229 2294 V 502 2298 2729 4 v 500 2397 4 100 v 564 2367
a(2)p 666 2397 V 123 w(2)p 831 2397 V 122 w(\()p Fs(P)12
b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 1313 2397
V 314 w(0)p 1859 2397 V 302 w(\()p Fs(P)12 b Fu(\))19
b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 2320 2397 V 283 w(2\()p
Fs(P)12 b Fu(\))18 b Fp(\000)g Fu(2\()p Fs(O)r Fu(\))p
3229 2397 V 502 2400 2729 4 v 500 2500 4 100 v 564 2470
a(2)p 666 2500 V 123 w(3)p 831 2500 V 127 w(\()p Fs(S)5
b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 1313 2500
V 128 w Fp(\000)p Fu(\()p Fs(T)12 b Fu(\))18 b(+)g(\()p
Fs(O)r Fu(\))p 1859 2500 V 114 w(\()p Fs(P)12 b Fu(\))19
b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 2320 2500 V 101 w(\()p
Fs(S)5 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(T)12 b Fu(\))18
b(+)g(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r
Fu(\))p 3229 2500 V 502 2503 2729 4 v 502 2520 V 500
2620 4 100 v 564 2590 a(3)p 666 2620 V 123 w(0)p 831
2620 V 282 w(0)p 1313 2620 V 285 w Fp(\000)p Fu(\()p
Fs(S)5 b Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p 1859 2620
V 120 w(\()p Fs(S)5 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r
Fu(\))p 2320 2620 V 489 w(0)p 3229 2620 V 502 2623 2729
4 v 500 2723 4 100 v 564 2693 a(3)p 666 2723 V 123 w(1)p
831 2723 V 127 w(\()p Fs(S)5 b Fu(\))19 b Fp(\000)f Fu(\()p
Fs(O)r Fu(\))p 1313 2723 V 318 w(0)p 1859 2723 V 306
w(\()p Fs(S)5 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p
2320 2723 V 293 w(2\()p Fs(S)5 b Fu(\))18 b Fp(\000)g
Fu(2\()p Fs(O)r Fu(\))p 3229 2723 V 502 2726 2729 4 v
500 2826 4 100 v 564 2796 a(3)p 666 2826 V 123 w(2)p
831 2826 V 122 w(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f
Fu(\()p Fs(O)r Fu(\))p 1313 2826 V 124 w Fp(\000)p Fu(\()p
Fs(T)12 b Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p 1859 2826
V 118 w(\()p Fs(S)5 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r
Fu(\))p 2320 2826 V 106 w(\()p Fs(P)12 b Fu(\))18 b Fp(\000)g
Fu(\()p Fs(T)12 b Fu(\))19 b(+)f(\()p Fs(S)5 b Fu(\))18
b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 3229 2826 V 502 2829
2729 4 v 500 2928 4 100 v 564 2899 a(3)p 666 2928 V 123
w(3)p 831 2928 V 125 w(\()p Fs(T)12 b Fu(\))18 b Fp(\000)g
Fu(\()p Fs(O)r Fu(\))p 1313 2928 V 124 w Fp(\000)p Fu(\()p
Fs(P)12 b Fu(\))19 b(+)f(\()p Fs(O)r Fu(\))p 1859 2928
V 115 w(\()p Fs(S)5 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r
Fu(\))p 2320 2928 V 106 w(\()p Fs(T)12 b Fu(\))18 b Fp(\000)g
Fu(\()p Fs(P)12 b Fu(\))19 b(+)f(\()p Fs(S)5 b Fu(\))18
b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 3229 2928 V 502 2932
2729 4 v 94 3159 a(Rewriting)26 b(this)h(table)f(in)h(more)f(compact)g
(form)g(w)n(e)h(obtain)g(the)h(statemen)n(t)e(of)i(the)g(prop)r
(osition.)p 3585 3109 57 4 v 3585 3159 4 50 v 3638 3159
V 3585 3162 57 4 v 94 3432 a Fk(4.3)113 b Fj(Cr)-5 b(oss)34
b(pr)-5 b(o)g(ducts)35 b(c)-5 b(orr)g(esp)g(onding)33
b(to)i(c)-5 b(o)g(cycles)34 b(of)h(typ)-5 b(e)35 b Fa(I)8
b(I)94 3585 y Fu(W)-7 b(e)28 b(need)g(some)e(lemma.)34
b(F)-7 b(or)27 b(short)g(w)n(e)g(set)1233 3752 y Fs(D)1302
3764 y Fo(P)1380 3752 y Fu(=)c(2\()p Fs(P)12 b Fu(\))18
b Fp(\000)g Fu(2\()p Fs(O)r Fu(\))24 b(=)f(div\()p Fs(x)c
Fp(\000)f Fs(x)2365 3764 y Fo(P)2421 3752 y Fu(\))23
b Fs(;)1239 3918 y(D)1308 3930 y Fo(T)1383 3918 y Fu(=)f(2\()p
Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(2\()p Fs(O)r Fu(\))24
b(=)e(div\()p Fs(x)e Fp(\000)e Fs(x)2363 3930 y Fo(T)2415
3918 y Fu(\))24 b Fs(;)1243 4058 y(D)1312 4070 y Fo(S)1383
4058 y Fu(=)e(2\()p Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(2\()p
Fs(O)r Fu(\))24 b(=)e(div\()p Fs(x)e Fp(\000)e Fs(x)2363
4070 y Fo(S)2411 4058 y Fu(\))24 b Fs(;)94 4198 y Fu(and)1327
4297 y Fs(D)h Fu(=)e(\()p Fs(P)12 b Fu(\))19 b(+)f(\()p
Fs(T)12 b Fu(\))18 b(+)g(\()p Fs(S)5 b Fu(\))18 b Fp(\000)g
Fu(3\()p Fs(O)r Fu(\))24 b Fs(:)94 4437 y Fu(Let)1481
4537 y(\000)f(=)f Fp(h)p Fs(D)r(;)14 b(D)1852 4549 y
Fo(P)1908 4537 y Fs(;)g(D)2014 4549 y Fo(T)2066 4537
y Fs(;)g(D)2172 4549 y Fo(S)2219 4537 y Fp(i)94 4676
y Fu(b)r(e)28 b(the)g(subgroup)f(in)808 4684 y(P)864
4676 y(\()916 4655 y(\026)896 4676 y Fs(E)6 b Fu(\))28
b(generated)e(b)n(y)h(these)h(principle)d(divisors.)33
b(Also)26 b(let)1299 4843 y(\012)d(=)g Fp(h)p Fs(y)s(;)14
b(x)k Fp(\000)g Fs(x)1778 4855 y Fo(P)1834 4843 y Fs(;)c(x)19
b Fp(\000)f Fs(x)2067 4855 y Fo(T)2120 4843 y Fs(;)c(x)19
b Fp(\000)f Fs(x)2353 4855 y Fo(S)2401 4843 y Fp(i)94
5009 y Fu(b)r(e)33 b(the)g(subgroup)f(in)829 4987 y(\026)827
5009 y Fs(k)s Fu(\()925 4988 y(\026)905 5009 y Fs(E)5
b Fu(\))1003 4979 y Fm(\003)1074 5009 y Fu(generated)32
b(b)n(y)g(the)h(functions)f Fs(x)22 b Fp(\000)f Fs(x)2291
5021 y Fo(P)2347 5009 y Fu(,)34 b Fs(x)22 b Fp(\000)f
Fs(x)2606 5021 y Fo(T)2659 5009 y Fu(,)34 b Fs(x)22 b
Fp(\000)f Fs(x)2918 5021 y Fo(S)2999 5009 y Fu(and)32
b Fs(y)s Fu(.)52 b(It)33 b(is)e(clear)94 5109 y(that)i(\000)f(and)g
(\012)g(are)f(discrete)f Fs(G)p Fu(-submo)r(dules)h(in)1723
5117 y(P)1779 5109 y(\()1831 5088 y(\026)1811 5109 y
Fs(E)6 b Fu(\))32 b(and)2110 5087 y(\026)2108 5109 y
Fs(k)s Fu(\()2205 5088 y(\026)2186 5109 y Fs(E)5 b Fu(\))2284
5079 y Fm(\003)2355 5109 y Fu(resp)r(ectiv)n(ely)-7 b(.)47
b(The)32 b(restriction)d(of)j(the)94 5209 y(homomorphism)23
b(div)g(:)866 5187 y(\026)864 5209 y Fs(k)s Fu(\()962
5188 y(\026)942 5209 y Fs(E)5 b Fu(\))1040 5178 y Fm(\003)1102
5209 y Fp(!)1208 5217 y Fu(P)1265 5209 y(\()1316 5188
y(\026)1297 5209 y Fs(E)g Fu(\))28 b(on)f(the)h(submo)r(dule)e(\000)i
(is)e(a)h(surjectiv)n(e)f Fs(G)p Fu(-homomorphism)1663
5423 y(\012)1801 5370 y Fn(div)1922 5395 y Fd(/)p Fc(/)f
Fd(/)p Fc(/)p 1747 5396 200 4 v 1972 5423 a Fu(\000)2072
5415 y Fs(:)1825 5697 y Fu(14)p eop
%%Page: 15 15
15 14 bop 94 98 a Fv(Lemma)30 b(4.3.1)40 b Ft(L)l(et)21
b Fs(F)35 b Fu(=)22 b Fs(F)12 b Fp(h)p Fs(D)r(;)i(D)1255
110 y Fo(P)1310 98 y Fs(;)g(D)1416 110 y Fo(T)1468 98
y Fs(;)g(D)1574 110 y Fo(S)1622 98 y Fp(i)22 b Ft(b)l(e)f(the)g(fr)l(e)
l(e)h(ab)l(elian)h(gr)l(oup)e(gener)l(ate)l(d)h(by)f(the)h(set)f
Fp(f)p Fs(D)r(;)14 b(D)3468 110 y Fo(P)3522 98 y Fs(;)g(D)3628
110 y Fo(T)3680 98 y Fs(;)g(D)3786 110 y Fo(S)3834 98
y Fp(g)p Ft(.)94 197 y(L)l(et)25 b Fs(\034)32 b Fu(:)23
b Fs(F)35 b Fp(!)23 b Fu(\012)i Ft(b)l(e)f(the)h(homomorphism)i
(de\014ne)l(d)e(by)g Fs(\034)9 b Fu(\()p Fs(D)1956 209
y Fo(P)2012 197 y Fu(\))23 b(=)g Fs(x)7 b Fp(\000)g Fs(x)2328
209 y Fo(P)2384 197 y Ft(,)25 b Fs(\034)9 b Fu(\()p Fs(D)2580
209 y Fo(T)2633 197 y Fu(\))24 b(=)e Fs(x)7 b Fp(\000)g
Fs(x)2949 209 y Fo(T)3002 197 y Ft(,)26 b Fs(\034)9 b
Fu(\()p Fs(D)3199 209 y Fo(S)3248 197 y Fu(\))23 b(=)g
Fs(x)7 b Fp(\000)g Fs(x)3564 209 y Fo(S)3612 197 y Ft(,)94
297 y Fs(\034)i Fu(\()p Fs(D)r Fu(\))25 b(=)d Fs(y)s
Ft(.)39 b(L)l(et)28 b Fs(\026)23 b Fu(:)g Fs(F)35 b Fp(!)23
b Fu(\000)29 b Ft(b)l(e)g(the)h(homomorphism)h(which)g(sends)e(any)h
(element)f(of)g(the)h(b)l(asis)f(of)h(the)f(gr)l(oup)94
397 y Fs(F)40 b Ft(to)27 b(itself.)39 b(Then)28 b(ther)l(e)f(exists)g
(a)h(unique)e(homomorphism)k(\(of)e(discr)l(ete)g Fs(G)p
Ft(-mo)l(dules\))g Fs(\015)5 b Ft(,)28 b(which)h(makes)f(the)94
496 y(diagr)l(am)1662 636 y Fs(F)1834 571 y Fo(\026)1956
607 y Fd(/)p Fc(/)p 1752 609 204 4 v 1789 810 a Fo(\034)1952
859 y Fd(\037)p Fc(\037)1922 830 y Fb(@)1894 802 y(@)1865
774 y(@)1836 745 y(@)1807 717 y(@)1778 689 y(@)1749 660
y(@)1981 636 y Fu(\000)2031 767 y Fo(\015)2006 859 y
Fd(\017)p Fc(\017)p 2005 859 4 200 v 1976 941 a Fu(\012)94
1082 y Ft(c)l(ommutative.)39 b(In)30 b(p)l(articular,)1622
1181 y Fu(div)14 b Fp(\016)p Fs(\015)27 b Fu(=)c(id)2019
1193 y Fn(\000)2087 1181 y Fs(:)94 1347 y Ft(Pr)l(o)l(of)p
Fu(.)55 b(Observ)n(e)31 b(that)i(2)p Fs(D)h Fu(=)d Fs(D)1183
1359 y Fo(P)1260 1347 y Fu(+)22 b Fs(D)1416 1359 y Fo(T)1490
1347 y Fu(+)g Fs(D)1646 1359 y Fo(S)1693 1347 y Fu(.)53
b(Sho)n(w)33 b(that)g(it)f(is)g(a)h(unique)f(relation)e(in)i(\000.)53
b(Indeed,)35 b(let)94 1447 y Fs(l)r(;)14 b(m;)g(n;)g(t)22
b Fp(2)i Fl(Z)d Fu(and)27 b Fs(D)21 b Fu(+)d Fs(mD)1045
1459 y Fo(P)1118 1447 y Fu(+)g Fs(nD)1320 1459 y Fo(T)1391
1447 y Fu(+)g Fs(tD)1573 1459 y Fo(S)1643 1447 y Fu(=)23
b(0.)37 b(Then)380 1630 y Fs(l)r Fu(\()p Fs(P)12 b Fu(\))18
b(+)g Fs(l)r Fu(\()p Fs(T)12 b Fu(\))18 b(+)g Fs(l)r
Fu(\()p Fs(S)5 b Fu(\))18 b Fp(\000)g Fu(3)p Fs(l)r Fu(\()p
Fs(O)r Fu(\))h(+)f(2)p Fs(m)p Fu(\()p Fs(P)12 b Fu(\))18
b Fp(\000)g Fu(2)p Fs(m)p Fu(\()p Fs(O)r Fu(\))h(+)f(2)p
Fs(n)p Fu(\()p Fs(T)12 b Fu(\))17 b Fp(\000)i Fu(2)p
Fs(n)p Fu(\()p Fs(O)r Fu(\))g(+)f(2)p Fs(t)p Fu(\()p
Fs(S)5 b Fu(\))18 b Fp(\000)g Fu(2)p Fs(t)p Fu(\()p Fs(O)r
Fu(\))24 b(=)525 1812 y(\()p Fs(l)c Fu(+)e(2)p Fs(m)p
Fu(\)\()p Fs(P)12 b Fu(\))18 b(+)g(\()p Fs(l)i Fu(+)e(2)p
Fs(n)p Fu(\)\()p Fs(T)12 b Fu(\))18 b(+)g(\()p Fs(l)i
Fu(+)e(2)p Fs(t)p Fu(\)\()p Fs(S)5 b Fu(\))19 b Fp(\000)f
Fu(2\()p Fs(m)g Fu(+)g Fs(n)g Fu(+)g Fs(t)p Fu(\)\()p
Fs(O)r Fu(\))i Fp(\000)e Fu(3)p Fs(l)r Fu(\()p Fs(O)r
Fu(\))23 b(=)g(0)g Fp(\))1338 2111 y(\))1444 1891 y Fi(8)1444
1966 y(>)1444 1991 y(>)1444 2016 y(<)1444 2165 y(>)1444
2190 y(>)1444 2215 y(:)1559 1961 y Fs(l)d Fu(+)e(2)p
Fs(m)23 b Fu(=)f(0)1559 2061 y Fs(l)e Fu(+)e(2)p Fs(n)23
b Fu(=)f(0)1559 2160 y Fs(l)e Fu(+)e(2)p Fs(t)23 b Fu(=)g(0)1559
2260 y(2\()p Fs(m)c Fu(+)f Fs(n)g Fu(+)g Fs(t)p Fu(\))h(+)f(3)p
Fs(l)24 b Fu(=)e(0)94 2406 y(Solving)28 b(this)i(system)g(one)g(can)g
(see)h(that)f Fs(m)e Fu(=)g Fs(n)g Fu(=)g Fs(t)j Fu(and)f
Fs(l)g Fu(=)d Fp(\000)p Fu(2)p Fs(m)p Fu(.)45 b(Hence)31
b(the)g(relation)d Fs(l)r(D)22 b Fu(+)e Fs(mD)3498 2418
y Fo(P)3573 2406 y Fu(+)94 2505 y Fs(nD)213 2517 y Fo(T)284
2505 y Fu(+)e Fs(tD)466 2517 y Fo(S)537 2505 y Fu(=)k(0)28
b(is)e(of)i(the)g(form)e Fp(\000)p Fu(2)p Fs(mD)19 b
Fu(+)f Fs(mD)1704 2517 y Fo(P)1778 2505 y Fu(+)g Fs(mD)2003
2517 y Fo(T)2073 2505 y Fu(+)g Fs(mD)2298 2517 y Fo(S)2369
2505 y Fu(=)23 b(0,)k(that)h(is)e(2)p Fs(D)f Fu(=)d Fs(D)3104
2517 y Fo(P)3178 2505 y Fu(+)c Fs(D)3330 2517 y Fo(T)3400
2505 y Fu(+)g Fs(D)3552 2517 y Fo(S)3600 2505 y Fu(.)219
2605 y(Th)n(us)31 b(w)n(e)f(ha)n(v)n(e)g(that)h(\000)f(is)g(the)h
(quotien)n(t)f(group)g(of)g Fs(F)41 b Fu(=)28 b Fs(F)12
b Fp(h)p Fs(D)r(;)i(D)2379 2617 y Fo(P)2434 2605 y Fs(;)g(D)2540
2617 y Fo(T)2592 2605 y Fs(;)g(D)2698 2617 y Fo(S)2746
2605 y Fp(i)31 b Fu(b)n(y)f(the)h(cyclic)e(subgroup)94
2704 y(generated)d(b)n(y)g(the)i(elemen)n(t)c Fs(D)1103
2716 y Fo(P)1175 2704 y Fu(+)16 b Fs(D)1325 2716 y Fo(T)1394
2704 y Fu(+)g Fs(D)1544 2716 y Fo(S)1609 2704 y Fp(\000)g
Fu(2)p Fs(D)r Fu(.)36 b(Then)27 b(the)g(existence)e(of)i
Fs(\015)k Fu(follo)n(ws)24 b(from)h(the)j(equalit)n(y)682
2936 y Fs(\034)9 b Fu(\()p Fs(D)828 2948 y Fo(P)903 2936
y Fu(+)18 b Fs(D)1055 2948 y Fo(T)1125 2936 y Fu(+)g
Fs(D)1277 2948 y Fo(S)1343 2936 y Fp(\000)g Fu(2)p Fs(D)r
Fu(\))23 b(=)1692 2879 y(\()p Fs(x)c Fp(\000)f Fs(x)1920
2891 y Fo(P)1976 2879 y Fu(\)\()p Fs(x)h Fp(\000)f Fs(x)2236
2891 y Fo(T)2289 2879 y Fu(\)\()p Fs(x)i Fp(\000)e Fs(x)2550
2891 y Fo(S)2598 2879 y Fu(\))p 1692 2917 939 4 v 2121
2993 a Fs(y)2165 2969 y Fn(2)2663 2936 y Fu(=)2761 2879
y Fs(y)2805 2849 y Fn(2)p 2761 2917 81 4 v 2761 2993
a Fs(y)2805 2969 y Fn(2)2875 2936 y Fu(=)23 b(1)f Fs(:)p
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568 y(Prop)r(osition)26 b(4.3.1)i(sho)n(ws)g(that)h(the)h(2-co)r(cycle)
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V 1659 2148 a Fo(y)p 1592 2166 169 4 v 1592 2214 a(x)p
Fm(\000)p Fo(x)1720 2222 y Fg(S)p 1851 2222 4 107 v 2009
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100 v 1358 2631 V 1500 2631 V 1851 2631 V 2199 2631 V
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2734 V 1500 2734 V 1851 2734 V 2199 2734 V 2543 2734
V 1185 2834 V 1253 2804 a Fu(1)p 1358 2834 V 115 w(1)p
1500 2834 V 204 w(1)p 1851 2834 V 308 w(1)p 2199 2834
V 304 w(1)p 2543 2834 V 1185 2933 V 1358 2933 V 1500
2933 V 1851 2933 V 2199 2933 V 2543 2933 V 1187 2937
1358 4 v 1185 3036 4 100 v 1358 3036 V 1500 3036 V 1851
3036 V 2199 3036 V 2543 3036 V 1185 3136 V 1358 3136
V 1410 3106 a(1)p 1500 3136 V 188 w Fs(c)1676 3076 y
Fn(2)p 1851 3136 V 2009 3106 a Fs(c)p 2199 3136 V 291
w(c)2372 3076 y Fn(3)p 2543 3136 V 1185 3235 V 1358 3235
V 1500 3235 V 1851 3235 V 2199 3235 V 2543 3235 V 1187
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3582 y Fs(k)s Fu(\()2712 3561 y(\026)2692 3582 y Fs(E)6
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Fm(\003)869 3980 y Fp(!)41 b Fs(k)s Fu(\()p Fs(E)5 b
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4080 y(isomorphism)1155 4196 y Fs(v)26 b Fu(:)d Fs(H)1343
4162 y Fn(2)1380 4196 y Fu(\()p Fs(C)q(;)14 b(Z)6 b Fu(\()p
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y Fu(=)1193 4950 y(=)1376 4894 y Fs(y)1420 4864 y Fn(2)p
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eop
%%Page: 18 18
18 17 bop 1462 18 809 4 v 1460 118 4 100 v 1512 88 a
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50 v 3638 1441 V 3585 1444 57 4 v 94 1756 a Fw(5)135
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Fs(;)14 b(g)1713 3533 y Fn(2)1750 3521 y Fu(\))23 b Fp(2)h
Fs(V)1932 3533 y Fo(P)2006 3521 y Fp(\002)18 b Fs(V)2137
3533 y Fo(P)990 3621 y Fu(div)q(\()1213 3584 y Fo(y)p
1147 3602 169 4 v 1147 3649 a(x)p Fm(\000)p Fo(x)1275
3657 y Fg(S)1326 3621 y Fu(\))23 b Fs(;)60 b Fu(\()p
Fs(g)1536 3633 y Fn(1)1573 3621 y Fs(;)14 b(g)1650 3633
y Fn(2)1687 3621 y Fu(\))23 b Fp(2)h Fu(\()p Fs(V)1901
3633 y Fo(T)1972 3621 y Fp(\002)18 b Fs(V)2103 3633 y
Fo(P)2159 3621 y Fu(\))h Fp([)g Fu(\()p Fs(U)2373 3633
y Fo(P)2446 3621 y Fp(\002)f Fs(V)2577 3633 y Fo(T)2630
3621 y Fu(\))h Fp([)g Fu(\()p Fs(g)2827 3633 y Fn(0)2864
3621 y Fs(U)2921 3633 y Fo(P)2994 3621 y Fp(\002)f Fs(V)3125
3633 y Fo(S)3174 3621 y Fu(\))990 3727 y(div)q(\()1216
3690 y Fo(y)p 1147 3708 174 4 v 1147 3756 a(x)p Fm(\000)p
Fo(x)1275 3764 y Fg(T)1330 3727 y Fu(\))23 b Fs(;)60
b Fu(\()p Fs(g)1540 3739 y Fn(1)1577 3727 y Fs(;)14 b(g)1654
3739 y Fn(2)1691 3727 y Fu(\))24 b Fp(2)f Fu(\()p Fs(V)1905
3739 y Fo(S)1972 3727 y Fp(\002)18 b Fs(V)2103 3739 y
Fo(P)2159 3727 y Fu(\))h Fp([)g Fu(\()p Fs(U)2373 3739
y Fo(P)2446 3727 y Fp(\002)f Fs(V)2577 3739 y Fo(S)2626
3727 y Fu(\))h Fp([)f Fu(\()p Fs(g)2822 3739 y Fn(0)2860
3727 y Fs(U)2917 3739 y Fo(P)2990 3727 y Fp(\002)g Fs(V)3121
3739 y Fo(T)3174 3727 y Fu(\))990 3834 y(div)q(\()1216
3797 y Fo(y)p 1147 3815 175 4 v 1147 3863 a(x)p Fm(\000)p
Fo(x)1275 3871 y Fg(P)1332 3834 y Fu(\))23 b Fs(;)60
b Fu(\()p Fs(g)1542 3846 y Fn(1)1579 3834 y Fs(;)14 b(g)1656
3846 y Fn(2)1693 3834 y Fu(\))23 b Fp(2)h Fu(\(\()p Fs(U)1948
3846 y Fo(S)2015 3834 y Fp([)18 b Fs(g)2128 3846 y Fn(0)2165
3834 y Fs(U)2222 3846 y Fo(S)2270 3834 y Fu(\))h Fp(\002)f
Fs(V)2452 3846 y Fo(T)2504 3834 y Fu(\))h Fp([)g Fu(\(\()p
Fs(U)2750 3846 y Fo(T)2821 3834 y Fp([)g Fs(g)2935 3846
y Fn(0)2972 3834 y Fs(U)3029 3846 y Fo(T)3081 3834 y
Fu(\))f Fp(\002)g Fs(V)3262 3846 y Fo(S)3311 3834 y Fu(\))94
4004 y Ft(Pr)l(o)l(of)p Fu(.)39 b(F)-7 b(or)27 b(the)h(pro)r(of)f(w)n
(e)g(use)g(the)h(form)n(ula)925 4170 y Fs(@)985 4148
y Fu(^)974 4170 y Fs(\036)1023 4182 y Fo(O)r(; )1145
4170 y Fu(\()p Fs(g)1217 4182 y Fn(1)1255 4170 y Fs(;)14
b(g)1332 4182 y Fn(2)1368 4170 y Fu(\))24 b(=)e Fs(g)1551
4182 y Fn(1)1599 4148 y Fu(^)1588 4170 y Fs(\036)1637
4182 y Fo(O)r(; )1760 4170 y Fu(\()p Fs(g)1832 4182 y
Fn(2)1869 4170 y Fu(\))d Fp(\000)2014 4148 y Fu(^)2003
4170 y Fs(\036)2052 4182 y Fo(O)r(; )2174 4170 y Fu(\()p
Fs(g)2246 4182 y Fn(1)2283 4170 y Fs(g)2323 4182 y Fn(2)2360
4170 y Fu(\))g(+)2505 4148 y(^)2494 4170 y Fs(\036)2543
4182 y Fo(O)r(; )2666 4170 y Fu(\()p Fs(g)2738 4182 y
Fn(1)2775 4170 y Fu(\))94 4336 y(and)28 b(the)g(rule)e(of)i(m)n
(ultiplication)21 b(in)27 b Fs(G=U)1453 4348 y Fo(O)1532
4314 y Fp(\030)1532 4340 y Fu(=)1620 4336 y Fs(D)1689
4348 y Fn(4)1726 4336 y Fu(.)37 b(All)26 b(calculations)d(are)k(in)g
(the)h(follo)n(wing)23 b(eigh)n(t)k(tables:)p 746 4530
2240 4 v 744 4629 4 100 v 814 4599 a Fs(g)854 4611 y
Fn(1)p 957 4629 V 1065 4599 a Fs(g)1105 4611 y Fn(2)p
1247 4629 V 1264 4629 V 1315 4599 a Fs(g)1355 4611 y
Fn(1)1392 4599 y Fs(\036)1441 4611 y Fo(O)r(; )1564 4599
y Fu(\()p Fs(g)1636 4611 y Fn(2)1673 4599 y Fu(\))p 1753
4629 V 100 w Fp(\000)p Fs(\036)1919 4611 y Fo(O)r(; )2041
4599 y Fu(\()p Fs(g)2113 4611 y Fn(1)2150 4599 y Fs(g)2190
4611 y Fn(2)2227 4599 y Fu(\))p 2307 4629 V 100 w Fs(\036)2408
4611 y Fo(O)r(; )2530 4599 y Fu(\()p Fs(g)2602 4611 y
Fn(1)2640 4599 y Fu(\))p 2720 4629 V 2737 4629 V 116
w(sum)p 2984 4629 V 746 4633 2240 4 v 746 4649 V 744
4749 4 100 v 796 4719 a Fs(U)853 4731 y Fo(O)p 957 4749
V 1047 4719 a Fs(U)1104 4731 y Fo(O)p 1247 4749 V 1264
4749 V 1489 4719 a Fu(0)p 1753 4749 V 480 w(0)p 2307
4749 V 442 w(0)p 2720 4749 V 2737 4749 V 304 w(0)p 2984
4749 V 746 4752 2240 4 v 744 4852 4 100 v 796 4822 a
Fs(U)853 4834 y Fo(O)p 957 4852 V 1052 4822 a Fs(U)1109
4834 y Fo(S)p 1247 4852 V 1264 4852 V 1334 4822 a Fu(\()p
Fs(S)5 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p
1753 4852 V 139 w Fp(\000)p Fu(\()p Fs(S)5 b Fu(\))18
b(+)g(\()p Fs(O)r Fu(\))p 2307 4852 V 256 w(0)p 2720
4852 V 2737 4852 V 304 w(0)p 2984 4852 V 746 4855 2240
4 v 744 4955 4 100 v 796 4925 a Fs(U)853 4937 y Fo(O)p
957 4955 V 1049 4925 a Fs(U)1106 4937 y Fo(T)p 1247 4955
V 1264 4955 V 1332 4925 a Fu(\()p Fs(T)12 b Fu(\))18
b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 1753 4955 V 134 w
Fp(\000)p Fu(\()p Fs(T)12 b Fu(\))18 b(+)g(\()p Fs(O)r
Fu(\))p 2307 4955 V 254 w(0)p 2720 4955 V 2737 4955 V
304 w(0)p 2984 4955 V 746 4958 2240 4 v 744 5058 4 100
v 796 5028 a Fs(U)853 5040 y Fo(O)p 957 5058 V 1048 5028
a Fs(U)1105 5040 y Fo(P)p 1247 5058 V 1264 5058 V 1330
5028 a Fu(\()p Fs(P)12 b Fu(\))18 b Fp(\000)g Fu(\()p
Fs(O)r Fu(\))p 1753 5058 V 130 w Fp(\000)p Fu(\()p Fs(P)12
b Fu(\))18 b(+)h(\()p Fs(O)r Fu(\))p 2307 5058 V 251
w(0)p 2720 5058 V 2737 5058 V 304 w(0)p 2984 5058 V 746
5061 2240 4 v 746 5078 V 744 5177 4 100 v 796 5147 a
Fs(U)853 5159 y Fo(O)p 957 5177 V 1009 5147 a Fs(g)1049
5159 y Fn(0)1086 5147 y Fs(U)1143 5159 y Fo(O)p 1247
5177 V 1264 5177 V 1489 5147 a Fu(0)p 1753 5177 V 480
w(0)p 2307 5177 V 442 w(0)p 2720 5177 V 2737 5177 V 304
w(0)p 2984 5177 V 746 5181 2240 4 v 744 5280 4 100 v
796 5250 a Fs(U)853 5262 y Fo(O)p 957 5280 V 1013 5250
a Fs(g)1053 5262 y Fn(0)1090 5250 y Fs(U)1147 5262 y
Fo(S)p 1247 5280 V 1264 5280 V 1332 5250 a Fu(\()p Fs(T)12
b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 1753 5280
V 134 w Fp(\000)p Fu(\()p Fs(T)12 b Fu(\))18 b(+)g(\()p
Fs(O)r Fu(\))p 2307 5280 V 254 w(0)p 2720 5280 V 2737
5280 V 304 w(0)p 2984 5280 V 746 5284 2240 4 v 744 5383
4 100 v 796 5353 a Fs(U)853 5365 y Fo(O)p 957 5383 V
1011 5353 a Fs(g)1051 5365 y Fn(0)1088 5353 y Fs(U)1145
5365 y Fo(T)p 1247 5383 V 1264 5383 V 1334 5353 a Fu(\()p
Fs(S)5 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p
1753 5383 V 139 w Fp(\000)p Fu(\()p Fs(S)5 b Fu(\))18
b(+)g(\()p Fs(O)r Fu(\))p 2307 5383 V 256 w(0)p 2720
5383 V 2737 5383 V 304 w(0)p 2984 5383 V 746 5386 2240
4 v 744 5486 4 100 v 796 5456 a Fs(U)853 5468 y Fo(O)p
957 5486 V 1009 5456 a Fs(g)1049 5468 y Fn(0)1086 5456
y Fs(U)1143 5468 y Fo(P)p 1247 5486 V 1264 5486 V 1330
5456 a Fu(\()p Fs(P)12 b Fu(\))18 b Fp(\000)g Fu(\()p
Fs(O)r Fu(\))p 1753 5486 V 130 w Fp(\000)p Fu(\()p Fs(P)12
b Fu(\))18 b(+)h(\()p Fs(O)r Fu(\))p 2307 5486 V 251
w(0)p 2720 5486 V 2737 5486 V 304 w(0)p 2984 5486 V 746
5489 2240 4 v 1825 5697 a(18)p eop
%%Page: 19 19
19 18 bop 401 18 2931 4 v 399 118 4 100 v 464 88 a Fs(g)504
100 y Fn(1)p 603 118 V 711 88 a Fs(g)751 100 y Fn(2)p
893 118 V 910 118 V 961 88 a Fs(g)1001 100 y Fn(1)1038
88 y Fs(\036)1087 100 y Fo(O)r(; )1210 88 y Fu(\()p Fs(g)1282
100 y Fn(2)1319 88 y Fu(\))p 1399 118 V 100 w Fp(\000)p
Fs(\036)1565 100 y Fo(O)r(; )1687 88 y Fu(\()p Fs(g)1759
100 y Fn(1)1796 88 y Fs(g)1836 100 y Fn(2)1873 88 y Fu(\))p
1953 118 V 119 w Fs(\036)2073 100 y Fo(O)r(; )2196 88
y Fu(\()p Fs(g)2268 100 y Fn(1)2305 88 y Fu(\))p 2405
118 V 2422 118 V 466 w(sum)p 3330 118 V 401 121 2931
4 v 401 138 V 399 237 4 100 v 450 207 a Fs(U)507 219
y Fo(S)p 603 237 V 693 207 a Fs(U)750 219 y Fo(O)p 893
237 V 910 237 V 1135 207 a Fu(0)p 1399 237 V 293 w Fp(\000)p
Fu(\()p Fs(S)5 b Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p 1953
237 V 120 w(\()p Fs(S)5 b Fu(\))18 b Fp(\000)g Fu(\()p
Fs(O)r Fu(\))p 2405 237 V 2422 237 V 502 w(0)p 3330 237
V 401 241 2931 4 v 399 340 4 100 v 450 310 a Fs(U)507
322 y Fo(S)p 603 340 V 697 310 a Fs(U)754 322 y Fo(S)p
893 340 V 910 340 V 980 310 a Fu(\()p Fs(S)5 b Fu(\))19
b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 1399 340 V 326 w(0)p
1953 340 V 306 w(\()p Fs(S)5 b Fu(\))18 b Fp(\000)g Fu(\()p
Fs(O)r Fu(\))p 2405 340 V 2422 340 V 305 w(2\()p Fs(S)5
b Fu(\))18 b Fp(\000)g Fu(2\()p Fs(O)r Fu(\))p 3330 340
V 401 344 2931 4 v 399 443 4 100 v 450 413 a Fs(U)507
425 y Fo(S)p 603 443 V 695 413 a Fs(U)752 425 y Fo(T)p
893 443 V 910 443 V 978 413 a Fu(\()p Fs(T)12 b Fu(\))18
b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 1399 443 V 132 w Fp(\000)p
Fu(\()p Fs(P)12 b Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p
1953 443 V 116 w(\()p Fs(S)5 b Fu(\))18 b Fp(\000)g Fu(\()p
Fs(O)r Fu(\))p 2405 443 V 2422 443 V 118 w(\()p Fs(T)12
b Fu(\))18 b Fp(\000)g Fu(\()p Fs(P)12 b Fu(\))19 b(+)f(\()p
Fs(S)5 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p
3330 443 V 401 447 2931 4 v 399 546 4 100 v 450 516 a
Fs(U)507 528 y Fo(S)p 603 546 V 694 516 a Fs(U)751 528
y Fo(P)p 893 546 V 910 546 V 975 516 a Fu(\()p Fs(P)12
b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 1399 546
V 132 w Fp(\000)p Fu(\()p Fs(T)12 b Fu(\))18 b(+)g(\()p
Fs(O)r Fu(\))p 1953 546 V 118 w(\()p Fs(S)5 b Fu(\))18
b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 2405 546 V 2422 546
V 118 w(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(T)12
b Fu(\))18 b(+)g(\()p Fs(S)5 b Fu(\))19 b Fp(\000)f Fu(\()p
Fs(O)r Fu(\))p 3330 546 V 401 549 2931 4 v 401 566 V
399 666 4 100 v 450 636 a Fs(U)507 648 y Fo(S)p 603 666
V 655 636 a Fs(g)695 648 y Fn(0)732 636 y Fs(U)789 648
y Fo(O)p 893 666 V 910 666 V 1135 636 a Fu(0)p 1399 666
V 293 w Fp(\000)p Fu(\()p Fs(S)5 b Fu(\))18 b(+)g(\()p
Fs(O)r Fu(\))p 1953 666 V 120 w(\()p Fs(S)5 b Fu(\))18
b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 2405 666 V 2422 666
V 502 w(0)p 3330 666 V 401 669 2931 4 v 399 769 4 100
v 450 739 a Fs(U)507 751 y Fo(S)p 603 769 V 659 739 a
Fs(g)699 751 y Fn(0)736 739 y Fs(U)793 751 y Fo(S)p 893
769 V 910 769 V 978 739 a Fu(\()p Fs(T)12 b Fu(\))18
b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 1399 769 V 132 w Fp(\000)p
Fu(\()p Fs(P)12 b Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p
1953 769 V 116 w(\()p Fs(S)5 b Fu(\))18 b Fp(\000)g Fu(\()p
Fs(O)r Fu(\))p 2405 769 V 2422 769 V 118 w(\()p Fs(T)12
b Fu(\))18 b Fp(\000)g Fu(\()p Fs(P)12 b Fu(\))19 b(+)f(\()p
Fs(S)5 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p
3330 769 V 401 772 2931 4 v 399 872 4 100 v 450 842 a
Fs(U)507 854 y Fo(S)p 603 872 V 657 842 a Fs(g)697 854
y Fn(0)734 842 y Fs(U)791 854 y Fo(T)p 893 872 V 910
872 V 980 842 a Fu(\()p Fs(S)5 b Fu(\))19 b Fp(\000)f
Fu(\()p Fs(O)r Fu(\))p 1399 872 V 326 w(0)p 1953 872
V 306 w(\()p Fs(S)5 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r
Fu(\))p 2405 872 V 2422 872 V 305 w(2\()p Fs(S)5 b Fu(\))18
b Fp(\000)g Fu(2\()p Fs(O)r Fu(\))p 3330 872 V 401 875
2931 4 v 399 975 4 100 v 450 945 a Fs(U)507 957 y Fo(S)p
603 975 V 655 945 a Fs(g)695 957 y Fn(0)732 945 y Fs(U)789
957 y Fo(P)p 893 975 V 910 975 V 975 945 a Fu(\()p Fs(P)12
b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 1399 975
V 132 w Fp(\000)p Fu(\()p Fs(T)12 b Fu(\))18 b(+)g(\()p
Fs(O)r Fu(\))p 1953 975 V 118 w(\()p Fs(S)5 b Fu(\))18
b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 2405 975 V 2422 975
V 118 w(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(T)12
b Fu(\))18 b(+)g(\()p Fs(S)5 b Fu(\))19 b Fp(\000)f Fu(\()p
Fs(O)r Fu(\))p 3330 975 V 401 978 2931 4 v 396 1084 2940
4 v 394 1184 4 100 v 462 1154 a Fs(g)502 1166 y Fn(1)p
603 1184 V 711 1154 a Fs(g)751 1166 y Fn(2)p 893 1184
V 910 1184 V 961 1154 a Fs(g)1001 1166 y Fn(1)1038 1154
y Fs(\036)1087 1166 y Fo(O)r(; )1209 1154 y Fu(\()p Fs(g)1281
1166 y Fn(2)1319 1154 y Fu(\))p 1399 1184 V 100 w Fp(\000)p
Fs(\036)1565 1166 y Fo(O)r(; )1687 1154 y Fu(\()p Fs(g)1759
1166 y Fn(1)1796 1154 y Fs(g)1836 1166 y Fn(2)1873 1154
y Fu(\))p 1953 1184 V 121 w Fs(\036)2075 1166 y Fo(O)r(; )2198
1154 y Fu(\()p Fs(g)2270 1166 y Fn(1)2307 1154 y Fu(\))p
2409 1184 V 2426 1184 V 469 w(sum)p 3334 1184 V 396 1187
2940 4 v 396 1204 V 394 1303 4 100 v 446 1273 a Fs(U)503
1285 y Fo(T)p 603 1303 V 693 1273 a Fs(U)750 1285 y Fo(O)p
893 1303 V 910 1303 V 1135 1273 a Fu(0)p 1399 1303 V
290 w Fp(\000)p Fu(\()p Fs(T)12 b Fu(\))18 b(+)g(\()p
Fs(O)r Fu(\))p 1953 1303 V 118 w(\()p Fs(T)12 b Fu(\))18
b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 2409 1303 V 2426 1303
V 501 w(0)p 3334 1303 V 396 1306 2940 4 v 394 1406 4
100 v 446 1376 a Fs(U)503 1388 y Fo(T)p 603 1406 V 697
1376 a Fs(U)754 1388 y Fo(S)p 893 1406 V 910 1406 V 980
1376 a Fu(\()p Fs(S)5 b Fu(\))19 b Fp(\000)f Fu(\()p
Fs(O)r Fu(\))p 1399 1406 V 134 w Fp(\000)p Fu(\()p Fs(P)12
b Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p 1953 1406 V 116
w(\()p Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r
Fu(\))p 2409 1406 V 2426 1406 V 117 w(\()p Fs(S)5 b Fu(\))19
b Fp(\000)f Fu(\()p Fs(P)12 b Fu(\))19 b(+)f(\()p Fs(T)12
b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 3334 1406
V 396 1409 2940 4 v 394 1509 4 100 v 446 1479 a Fs(U)503
1491 y Fo(T)p 603 1509 V 695 1479 a Fs(U)752 1491 y Fo(T)p
893 1509 V 910 1509 V 978 1479 a Fu(\()p Fs(T)12 b Fu(\))18
b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 1399 1509 V 324 w(0)p
1953 1509 V 306 w(\()p Fs(T)12 b Fu(\))18 b Fp(\000)g
Fu(\()p Fs(O)r Fu(\))p 2409 1509 V 2426 1509 V 302 w(2\()p
Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(2\()p Fs(O)r Fu(\))p
3334 1509 V 396 1512 2940 4 v 394 1612 4 100 v 446 1582
a Fs(U)503 1594 y Fo(T)p 603 1612 V 694 1582 a Fs(U)751
1594 y Fo(P)p 893 1612 V 910 1612 V 975 1582 a Fu(\()p
Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p
1399 1612 V 135 w Fp(\000)p Fu(\()p Fs(S)5 b Fu(\))18
b(+)g(\()p Fs(O)r Fu(\))p 1953 1612 V 120 w(\()p Fs(T)12
b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 2409 1612
V 2426 1612 V 117 w(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f
Fu(\()p Fs(S)5 b Fu(\))19 b(+)f(\()p Fs(T)12 b Fu(\))18
b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 3334 1612 V 396 1615
2940 4 v 396 1632 V 394 1732 4 100 v 446 1702 a Fs(U)503
1714 y Fo(T)p 603 1732 V 655 1702 a Fs(g)695 1714 y Fn(0)732
1702 y Fs(U)789 1714 y Fo(O)p 893 1732 V 910 1732 V 1135
1702 a Fu(0)p 1399 1732 V 290 w Fp(\000)p Fu(\()p Fs(T)12
b Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p 1953 1732 V 118
w(\()p Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r
Fu(\))p 2409 1732 V 2426 1732 V 501 w(0)p 3334 1732 V
396 1735 2940 4 v 394 1835 4 100 v 446 1805 a Fs(U)503
1817 y Fo(T)p 603 1835 V 659 1805 a Fs(g)699 1817 y Fn(0)736
1805 y Fs(U)793 1817 y Fo(S)p 893 1835 V 910 1835 V 978
1805 a Fu(\()p Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(\()p
Fs(O)r Fu(\))p 1399 1835 V 324 w(0)p 1953 1835 V 306
w(\()p Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r
Fu(\))p 2409 1835 V 2426 1835 V 302 w(2\()p Fs(T)12 b
Fu(\))18 b Fp(\000)g Fu(2\()p Fs(O)r Fu(\))p 3334 1835
V 396 1838 2940 4 v 394 1937 4 100 v 446 1908 a Fs(U)503
1920 y Fo(T)p 603 1937 V 657 1908 a Fs(g)697 1920 y Fn(0)734
1908 y Fs(U)791 1920 y Fo(T)p 893 1937 V 910 1937 V 980
1908 a Fu(\()p Fs(S)5 b Fu(\))19 b Fp(\000)f Fu(\()p
Fs(O)r Fu(\))p 1399 1937 V 134 w Fp(\000)p Fu(\()p Fs(P)12
b Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p 1953 1937 V 116
w(\()p Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r
Fu(\))p 2409 1937 V 2426 1937 V 117 w(\()p Fs(S)5 b Fu(\))19
b Fp(\000)f Fu(\()p Fs(P)12 b Fu(\))19 b(+)f(\()p Fs(T)12
b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 3334 1937
V 396 1941 2940 4 v 394 2040 4 100 v 446 2011 a Fs(U)503
2023 y Fo(T)p 603 2040 V 655 2011 a Fs(g)695 2023 y Fn(0)732
2011 y Fs(U)789 2023 y Fo(P)p 893 2040 V 910 2040 V 975
2011 a Fu(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p
Fs(O)r Fu(\))p 1399 2040 V 135 w Fp(\000)p Fu(\()p Fs(S)5
b Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p 1953 2040 V 120
w(\()p Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r
Fu(\))p 2409 2040 V 2426 2040 V 117 w(\()p Fs(P)12 b
Fu(\))19 b Fp(\000)f Fu(\()p Fs(S)5 b Fu(\))19 b(+)f(\()p
Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p
3334 2040 V 396 2044 2940 4 v 392 2150 2948 4 v 390 2249
4 100 v 460 2220 a Fs(g)500 2232 y Fn(1)p 602 2249 V
711 2220 a Fs(g)751 2232 y Fn(2)p 892 2249 V 909 2249
V 960 2220 a Fs(g)1000 2232 y Fn(1)1037 2220 y Fs(\036)1086
2232 y Fo(O)r(; )1209 2220 y Fu(\()p Fs(g)1281 2232 y
Fn(2)1318 2220 y Fu(\))p 1398 2249 V 100 w Fp(\000)p
Fs(\036)1564 2232 y Fo(O)r(; )1686 2220 y Fu(\()p Fs(g)1758
2232 y Fn(1)1795 2220 y Fs(g)1835 2232 y Fn(2)1872 2220
y Fu(\))p 1952 2249 V 124 w Fs(\036)2077 2232 y Fo(O)r(; )2199
2220 y Fu(\()p Fs(g)2271 2232 y Fn(1)2309 2220 y Fu(\))p
2413 2249 V 2430 2249 V 471 w(sum)p 3338 2249 V 392 2253
2948 4 v 392 2269 V 390 2369 4 100 v 442 2339 a Fs(U)499
2351 y Fo(P)p 602 2369 V 692 2339 a Fs(U)749 2351 y Fo(O)p
892 2369 V 909 2369 V 1134 2339 a Fu(0)p 1398 2369 V
288 w Fp(\000)p Fu(\()p Fs(P)12 b Fu(\))19 b(+)f(\()p
Fs(O)r Fu(\))p 1952 2369 V 115 w(\()p Fs(P)12 b Fu(\))19
b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 2413 2369 V 2430 2369
V 501 w(0)p 3338 2369 V 392 2372 2948 4 v 390 2472 4
100 v 442 2442 a Fs(U)499 2454 y Fo(P)p 602 2472 V 697
2442 a Fs(U)754 2454 y Fo(S)p 892 2472 V 909 2472 V 979
2442 a Fu(\()p Fs(S)5 b Fu(\))19 b Fp(\000)f Fu(\()p
Fs(O)r Fu(\))p 1398 2472 V 136 w Fp(\000)p Fu(\()p Fs(T)12
b Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p 1952 2472 V 118
w(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r
Fu(\))p 2413 2472 V 2430 2472 V 117 w(\()p Fs(S)5 b Fu(\))19
b Fp(\000)f Fu(\()p Fs(T)12 b Fu(\))18 b(+)g(\()p Fs(P)12
b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 3338 2472
V 392 2475 2948 4 v 390 2575 4 100 v 442 2545 a Fs(U)499
2557 y Fo(P)p 602 2575 V 694 2545 a Fs(U)751 2557 y Fo(T)p
892 2575 V 909 2575 V 977 2545 a Fu(\()p Fs(T)12 b Fu(\))18
b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 1398 2575 V 137 w
Fp(\000)p Fu(\()p Fs(S)5 b Fu(\))18 b(+)g(\()p Fs(O)r
Fu(\))p 1952 2575 V 120 w(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f
Fu(\()p Fs(O)r Fu(\))p 2413 2575 V 2430 2575 V 117 w(\()p
Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(S)5 b Fu(\))19
b(+)f(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r
Fu(\))p 3338 2575 V 392 2578 2948 4 v 390 2678 4 100
v 442 2648 a Fs(U)499 2660 y Fo(P)p 602 2678 V 693 2648
a Fs(U)750 2660 y Fo(P)p 892 2678 V 909 2678 V 975 2648
a Fu(\()p Fs(P)12 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r
Fu(\))p 1398 2678 V 322 w(0)p 1952 2678 V 306 w(\()p
Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p
2413 2678 V 2430 2678 V 299 w(2\()p Fs(P)12 b Fu(\))19
b Fp(\000)f Fu(2\()p Fs(O)r Fu(\))p 3338 2678 V 392 2681
2948 4 v 392 2698 V 390 2797 4 100 v 442 2768 a Fs(U)499
2780 y Fo(P)p 602 2797 V 654 2768 a Fs(g)694 2780 y Fn(0)731
2768 y Fs(U)788 2780 y Fo(O)p 892 2797 V 909 2797 V 1134
2768 a Fu(0)p 1398 2797 V 288 w Fp(\000)p Fu(\()p Fs(P)12
b Fu(\))19 b(+)f(\()p Fs(O)r Fu(\))p 1952 2797 V 115
w(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r
Fu(\))p 2413 2797 V 2430 2797 V 501 w(0)p 3338 2797 V
392 2801 2948 4 v 390 2900 4 100 v 442 2870 a Fs(U)499
2882 y Fo(P)p 602 2900 V 658 2870 a Fs(g)698 2882 y Fn(0)735
2870 y Fs(U)792 2882 y Fo(S)p 892 2900 V 909 2900 V 977
2870 a Fu(\()p Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(\()p
Fs(O)r Fu(\))p 1398 2900 V 137 w Fp(\000)p Fu(\()p Fs(S)5
b Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p 1952 2900 V 120
w(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r
Fu(\))p 2413 2900 V 2430 2900 V 117 w(\()p Fs(T)12 b
Fu(\))18 b Fp(\000)g Fu(\()p Fs(S)5 b Fu(\))19 b(+)f(\()p
Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p
3338 2900 V 392 2904 2948 4 v 390 3003 4 100 v 442 2973
a Fs(U)499 2985 y Fo(P)p 602 3003 V 656 2973 a Fs(g)696
2985 y Fn(0)733 2973 y Fs(U)790 2985 y Fo(T)p 892 3003
V 909 3003 V 979 2973 a Fu(\()p Fs(S)5 b Fu(\))19 b Fp(\000)f
Fu(\()p Fs(O)r Fu(\))p 1398 3003 V 136 w Fp(\000)p Fu(\()p
Fs(T)12 b Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p 1952 3003
V 118 w(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r
Fu(\))p 2413 3003 V 2430 3003 V 117 w(\()p Fs(S)5 b Fu(\))19
b Fp(\000)f Fu(\()p Fs(T)12 b Fu(\))18 b(+)g(\()p Fs(P)12
b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 3338 3003
V 392 3007 2948 4 v 390 3106 4 100 v 442 3076 a Fs(U)499
3088 y Fo(P)p 602 3106 V 654 3076 a Fs(g)694 3088 y Fn(0)731
3076 y Fs(U)788 3088 y Fo(P)p 892 3106 V 909 3106 V 975
3076 a Fu(\()p Fs(P)12 b Fu(\))18 b Fp(\000)g Fu(\()p
Fs(O)r Fu(\))p 1398 3106 V 322 w(0)p 1952 3106 V 306
w(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r
Fu(\))p 2413 3106 V 2430 3106 V 299 w(2\()p Fs(P)12 b
Fu(\))19 b Fp(\000)f Fu(2\()p Fs(O)r Fu(\))p 3338 3106
V 392 3110 2948 4 v 708 3216 2317 4 v 706 3315 4 100
v 814 3285 a Fs(g)854 3297 y Fn(1)p 996 3315 V 1104 3285
a Fs(g)1144 3297 y Fn(2)p 1285 3315 V 1302 3315 V 1354
3285 a Fs(g)1394 3297 y Fn(1)1430 3285 y Fs(\036)1479
3297 y Fo(O)r(; )1602 3285 y Fu(\()p Fs(g)1674 3297 y
Fn(2)1711 3285 y Fu(\))p 1792 3315 V 100 w Fp(\000)p
Fs(\036)1957 3297 y Fo(O)r(; )2079 3285 y Fu(\()p Fs(g)2151
3297 y Fn(1)2188 3285 y Fs(g)2228 3297 y Fn(2)2265 3285
y Fu(\))p 2346 3315 V 100 w Fs(\036)2446 3297 y Fo(O)r(; )2569
3285 y Fu(\()p Fs(g)2641 3297 y Fn(1)2678 3285 y Fu(\))p
2758 3315 V 2775 3315 V 116 w(sum)p 3023 3315 V 708 3319
2317 4 v 708 3335 V 706 3435 4 100 v 758 3405 a Fs(g)798
3417 y Fn(0)835 3405 y Fs(U)892 3417 y Fo(O)p 996 3435
V 1086 3405 a Fs(U)1143 3417 y Fo(O)p 1285 3435 V 1302
3435 V 1528 3405 a Fu(0)p 1792 3435 V 480 w(0)p 2346
3435 V 441 w(0)p 2758 3435 V 2775 3435 V 305 w(0)p 3023
3435 V 708 3438 2317 4 v 706 3538 4 100 v 758 3508 a
Fs(g)798 3520 y Fn(0)835 3508 y Fs(U)892 3520 y Fo(O)p
996 3538 V 1090 3508 a Fs(U)1147 3520 y Fo(S)p 1285 3538
V 1302 3538 V 1370 3508 a Fu(\()p Fs(T)12 b Fu(\))18
b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 1792 3538 V 135 w
Fp(\000)p Fu(\()p Fs(T)12 b Fu(\))18 b(+)g(\()p Fs(O)r
Fu(\))p 2346 3538 V 253 w(0)p 2758 3538 V 2775 3538 V
305 w(0)p 3023 3538 V 708 3541 2317 4 v 706 3641 4 100
v 758 3611 a Fs(g)798 3623 y Fn(0)835 3611 y Fs(U)892
3623 y Fo(O)p 996 3641 V 1088 3611 a Fs(U)1145 3623 y
Fo(T)p 1285 3641 V 1302 3641 V 1373 3611 a Fu(\()p Fs(S)5
b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 1792 3641
V 139 w Fp(\000)p Fu(\()p Fs(S)5 b Fu(\))18 b(+)g(\()p
Fs(O)r Fu(\))p 2346 3641 V 256 w(0)p 2758 3641 V 2775
3641 V 305 w(0)p 3023 3641 V 708 3644 2317 4 v 706 3744
4 100 v 758 3714 a Fs(g)798 3726 y Fn(0)835 3714 y Fs(U)892
3726 y Fo(O)p 996 3744 V 1086 3714 a Fs(U)1143 3726 y
Fo(P)p 1285 3744 V 1302 3744 V 1368 3714 a Fu(\()p Fs(P)12
b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 1792 3744
V 129 w Fp(\000)p Fu(\()p Fs(P)12 b Fu(\))19 b(+)f(\()p
Fs(O)r Fu(\))p 2346 3744 V 251 w(0)p 2758 3744 V 2775
3744 V 305 w(0)p 3023 3744 V 708 3747 2317 4 v 708 3764
V 706 3863 4 100 v 758 3833 a Fs(g)798 3845 y Fn(0)835
3833 y Fs(U)892 3845 y Fo(O)p 996 3863 V 1047 3833 a
Fs(g)1087 3845 y Fn(0)1124 3833 y Fs(U)1181 3845 y Fo(O)p
1285 3863 V 1302 3863 V 1528 3833 a Fu(0)p 1792 3863
V 480 w(0)p 2346 3863 V 441 w(0)p 2758 3863 V 2775 3863
V 305 w(0)p 3023 3863 V 708 3867 2317 4 v 706 3966 4
100 v 758 3936 a Fs(g)798 3948 y Fn(0)835 3936 y Fs(U)892
3948 y Fo(O)p 996 3966 V 1052 3936 a Fs(g)1092 3948 y
Fn(0)1128 3936 y Fs(U)1185 3948 y Fo(S)p 1285 3966 V
1302 3966 V 1373 3936 a Fu(\()p Fs(S)5 b Fu(\))18 b Fp(\000)g
Fu(\()p Fs(O)r Fu(\))p 1792 3966 V 139 w Fp(\000)p Fu(\()p
Fs(S)5 b Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p 2346 3966
V 256 w(0)p 2758 3966 V 2775 3966 V 305 w(0)p 3023 3966
V 708 3970 2317 4 v 706 4069 4 100 v 758 4039 a Fs(g)798
4051 y Fn(0)835 4039 y Fs(U)892 4051 y Fo(O)p 996 4069
V 1049 4039 a Fs(g)1089 4051 y Fn(0)1126 4039 y Fs(U)1183
4051 y Fo(T)p 1285 4069 V 1302 4069 V 1370 4039 a Fu(\()p
Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p
1792 4069 V 135 w Fp(\000)p Fu(\()p Fs(T)12 b Fu(\))18
b(+)g(\()p Fs(O)r Fu(\))p 2346 4069 V 253 w(0)p 2758
4069 V 2775 4069 V 305 w(0)p 3023 4069 V 708 4072 2317
4 v 706 4172 4 100 v 758 4142 a Fs(g)798 4154 y Fn(0)835
4142 y Fs(U)892 4154 y Fo(O)p 996 4172 V 1048 4142 a
Fs(g)1088 4154 y Fn(0)1125 4142 y Fs(U)1182 4154 y Fo(P)p
1285 4172 V 1302 4172 V 1368 4142 a Fu(\()p Fs(P)12 b
Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 1792 4172
V 129 w Fp(\000)p Fu(\()p Fs(P)12 b Fu(\))19 b(+)f(\()p
Fs(O)r Fu(\))p 2346 4172 V 251 w(0)p 2758 4172 V 2775
4172 V 305 w(0)p 3023 4172 V 708 4175 2317 4 v 360 4282
3013 4 v 358 4381 4 100 v 462 4351 a Fs(g)502 4363 y
Fn(1)p 640 4381 V 748 4351 a Fs(g)788 4363 y Fn(2)p 929
4381 V 946 4381 V 997 4351 a Fs(g)1037 4363 y Fn(1)1074
4351 y Fs(\036)1123 4363 y Fo(O)r(; )1246 4351 y Fu(\()p
Fs(g)1318 4363 y Fn(2)1355 4351 y Fu(\))p 1435 4381 V
100 w Fp(\000)p Fs(\036)1601 4363 y Fo(O)r(; )1723 4351
y Fu(\()p Fs(g)1795 4363 y Fn(1)1832 4351 y Fs(g)1872
4363 y Fn(2)1909 4351 y Fu(\))p 1989 4381 V 122 w Fs(\036)2112
4363 y Fo(O)r(; )2234 4351 y Fu(\()p Fs(g)2306 4363 y
Fn(1)2343 4351 y Fu(\))p 2445 4381 V 2462 4381 V 469
w(sum)p 3371 4381 V 360 4384 3013 4 v 360 4401 V 358
4501 4 100 v 410 4471 a Fs(g)450 4483 y Fn(0)487 4471
y Fs(U)544 4483 y Fo(S)p 640 4501 V 730 4471 a Fs(U)787
4483 y Fo(O)p 929 4501 V 946 4501 V 1172 4471 a Fu(0)p
1435 4501 V 290 w Fp(\000)p Fu(\()p Fs(T)12 b Fu(\))17
b(+)h(\()p Fs(O)r Fu(\))p 1989 4501 V 118 w(\()p Fs(T)12
b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 2445 4501
V 2462 4501 V 501 w(0)p 3371 4501 V 360 4504 3013 4 v
358 4604 4 100 v 410 4574 a Fs(g)450 4586 y Fn(0)487
4574 y Fs(U)544 4586 y Fo(S)p 640 4604 V 734 4574 a Fs(U)791
4586 y Fo(S)p 929 4604 V 946 4604 V 1014 4574 a Fu(\()p
Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p
1435 4604 V 324 w(0)p 1989 4604 V 306 w(\()p Fs(T)12
b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 2445 4604
V 2462 4604 V 302 w(2\()p Fs(T)12 b Fu(\))18 b Fp(\000)g
Fu(2\()p Fs(O)r Fu(\))p 3371 4604 V 360 4607 3013 4 v
358 4707 4 100 v 410 4677 a Fs(g)450 4689 y Fn(0)487
4677 y Fs(U)544 4689 y Fo(S)p 640 4707 V 732 4677 a Fs(U)789
4689 y Fo(T)p 929 4707 V 946 4707 V 1016 4677 a Fu(\()p
Fs(S)5 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p
1435 4707 V 134 w Fp(\000)p Fu(\()p Fs(P)12 b Fu(\))19
b(+)f(\()p Fs(O)r Fu(\))p 1989 4707 V 115 w(\()p Fs(T)12
b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 2445 4707
V 2462 4707 V 117 w(\()p Fs(S)5 b Fu(\))19 b Fp(\000)f
Fu(\()p Fs(P)12 b Fu(\))19 b(+)f(\()p Fs(T)12 b Fu(\))18
b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 3371 4707 V 360 4710
3013 4 v 358 4810 4 100 v 410 4780 a Fs(g)450 4792 y
Fn(0)487 4780 y Fs(U)544 4792 y Fo(S)p 640 4810 V 730
4780 a Fs(U)787 4792 y Fo(P)p 929 4810 V 946 4810 V 1012
4780 a Fu(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p
Fs(O)r Fu(\))p 1435 4810 V 134 w Fp(\000)p Fu(\()p Fs(S)5
b Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p 1989 4810 V 120
w(\()p Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r
Fu(\))p 2445 4810 V 2462 4810 V 117 w(\()p Fs(P)12 b
Fu(\))19 b Fp(\000)f Fu(\()p Fs(S)5 b Fu(\))19 b(+)f(\()p
Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p
3371 4810 V 360 4813 3013 4 v 360 4829 V 358 4929 4 100
v 410 4899 a Fs(g)450 4911 y Fn(0)487 4899 y Fs(U)544
4911 y Fo(S)p 640 4929 V 691 4899 a Fs(g)731 4911 y Fn(0)768
4899 y Fs(U)825 4911 y Fo(O)p 929 4929 V 946 4929 V 1172
4899 a Fu(0)p 1435 4929 V 290 w Fp(\000)p Fu(\()p Fs(T)12
b Fu(\))17 b(+)h(\()p Fs(O)r Fu(\))p 1989 4929 V 118
w(\()p Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r
Fu(\))p 2445 4929 V 2462 4929 V 501 w(0)p 3371 4929 V
360 4932 3013 4 v 358 5032 4 100 v 410 5002 a Fs(g)450
5014 y Fn(0)487 5002 y Fs(U)544 5014 y Fo(S)p 640 5032
V 695 5002 a Fs(g)735 5014 y Fn(0)772 5002 y Fs(U)829
5014 y Fo(S)p 929 5032 V 946 5032 V 1016 5002 a Fu(\()p
Fs(S)5 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p
1435 5032 V 134 w Fp(\000)p Fu(\()p Fs(P)12 b Fu(\))19
b(+)f(\()p Fs(O)r Fu(\))p 1989 5032 V 115 w(\()p Fs(T)12
b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 2445 5032
V 2462 5032 V 117 w(\()p Fs(S)5 b Fu(\))19 b Fp(\000)f
Fu(\()p Fs(P)12 b Fu(\))19 b(+)f(\()p Fs(T)12 b Fu(\))18
b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 3371 5032 V 360 5035
3013 4 v 358 5135 4 100 v 410 5105 a Fs(g)450 5117 y
Fn(0)487 5105 y Fs(U)544 5117 y Fo(S)p 640 5135 V 693
5105 a Fs(g)733 5117 y Fn(0)770 5105 y Fs(U)827 5117
y Fo(T)p 929 5135 V 946 5135 V 1014 5105 a Fu(\()p Fs(T)12
b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 1435 5135
V 324 w(0)p 1989 5135 V 306 w(\()p Fs(T)12 b Fu(\))18
b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 2445 5135 V 2462 5135
V 302 w(2\()p Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(2\()p
Fs(O)r Fu(\))p 3371 5135 V 360 5138 3013 4 v 358 5238
4 100 v 410 5208 a Fs(g)450 5220 y Fn(0)487 5208 y Fs(U)544
5220 y Fo(S)p 640 5238 V 692 5208 a Fs(g)732 5220 y Fn(0)769
5208 y Fs(U)826 5220 y Fo(P)p 929 5238 V 946 5238 V 1012
5208 a Fu(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p
Fs(O)r Fu(\))p 1435 5238 V 134 w Fp(\000)p Fu(\()p Fs(S)5
b Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p 1989 5238 V 120
w(\()p Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r
Fu(\))p 2445 5238 V 2462 5238 V 117 w(\()p Fs(P)12 b
Fu(\))19 b Fp(\000)f Fu(\()p Fs(S)5 b Fu(\))19 b(+)f(\()p
Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p
3371 5238 V 360 5241 3013 4 v 1825 5697 a(19)p eop
%%Page: 20 20
20 19 bop 360 18 3012 4 v 358 118 4 100 v 464 88 a Fs(g)504
100 y Fn(1)p 644 118 V 752 88 a Fs(g)792 100 y Fn(2)p
934 118 V 950 118 V 1002 88 a Fs(g)1042 100 y Fn(1)1078
88 y Fs(\036)1127 100 y Fo(O)r(; )1250 88 y Fu(\()p Fs(g)1322
100 y Fn(2)1359 88 y Fu(\))p 1440 118 V 100 w Fp(\000)p
Fs(\036)1605 100 y Fo(O)r(; )1727 88 y Fu(\()p Fs(g)1799
100 y Fn(1)1836 88 y Fs(g)1876 100 y Fn(2)1913 88 y Fu(\))p
1994 118 V 120 w Fs(\036)2114 100 y Fo(O)r(; )2236 88
y Fu(\()p Fs(g)2308 100 y Fn(1)2345 88 y Fu(\))p 2445
118 V 2462 118 V 467 w(sum)p 3370 118 V 360 121 3012
4 v 360 138 V 358 237 4 100 v 410 207 a Fs(g)450 219
y Fn(0)487 207 y Fs(U)544 219 y Fo(T)p 644 237 V 734
207 a Fs(U)791 219 y Fo(O)p 934 237 V 950 237 V 1176
207 a Fu(0)p 1440 237 V 292 w Fp(\000)p Fu(\()p Fs(S)5
b Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p 1994 237 V 120 w(\()p
Fs(S)5 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p
2445 237 V 2462 237 V 501 w(0)p 3370 237 V 360 241 3012
4 v 358 340 4 100 v 410 310 a Fs(g)450 322 y Fn(0)487
310 y Fs(U)544 322 y Fo(T)p 644 340 V 738 310 a Fs(U)795
322 y Fo(S)p 934 340 V 950 340 V 1018 310 a Fu(\()p Fs(T)12
b Fu(\))18 b Fp(\000)h Fu(\()p Fs(O)r Fu(\))p 1440 340
V 132 w Fp(\000)p Fu(\()p Fs(P)12 b Fu(\))18 b(+)g(\()p
Fs(O)r Fu(\))p 1994 340 V 115 w(\()p Fs(S)5 b Fu(\))19
b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 2445 340 V 2462 340
V 117 w(\()p Fs(T)12 b Fu(\))18 b Fp(\000)h Fu(\()p Fs(P)12
b Fu(\))18 b(+)g(\()p Fs(S)5 b Fu(\))19 b Fp(\000)f Fu(\()p
Fs(O)r Fu(\))p 3370 340 V 360 344 3012 4 v 358 443 4
100 v 410 413 a Fs(g)450 425 y Fn(0)487 413 y Fs(U)544
425 y Fo(T)p 644 443 V 736 413 a Fs(U)793 425 y Fo(T)p
934 443 V 950 443 V 1021 413 a Fu(\()p Fs(S)5 b Fu(\))18
b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 1440 443 V 327 w(0)p
1994 443 V 305 w(\()p Fs(S)5 b Fu(\))19 b Fp(\000)f Fu(\()p
Fs(O)r Fu(\))p 2445 443 V 2462 443 V 304 w(2\()p Fs(S)5
b Fu(\))19 b Fp(\000)f Fu(2\()p Fs(O)r Fu(\))p 3370 443
V 360 447 3012 4 v 358 546 4 100 v 410 516 a Fs(g)450
528 y Fn(0)487 516 y Fs(U)544 528 y Fo(T)p 644 546 V
734 516 a Fs(U)791 528 y Fo(P)p 934 546 V 950 546 V 1016
516 a Fu(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p
Fs(O)r Fu(\))p 1440 546 V 132 w Fp(\000)p Fu(\()p Fs(T)12
b Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p 1994 546 V 117 w(\()p
Fs(S)5 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p
2445 546 V 2462 546 V 117 w(\()p Fs(P)12 b Fu(\))19 b
Fp(\000)f Fu(\()p Fs(T)12 b Fu(\))18 b(+)g(\()p Fs(S)5
b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 3370 546
V 360 549 3012 4 v 360 566 V 358 666 4 100 v 410 636
a Fs(g)450 648 y Fn(0)487 636 y Fs(U)544 648 y Fo(T)p
644 666 V 695 636 a Fs(g)735 648 y Fn(0)772 636 y Fs(U)829
648 y Fo(O)p 934 666 V 950 666 V 1176 636 a Fu(0)p 1440
666 V 292 w Fp(\000)p Fu(\()p Fs(S)5 b Fu(\))18 b(+)g(\()p
Fs(O)r Fu(\))p 1994 666 V 120 w(\()p Fs(S)5 b Fu(\))19
b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 2445 666 V 2462 666
V 501 w(0)p 3370 666 V 360 669 3012 4 v 358 769 4 100
v 410 739 a Fs(g)450 751 y Fn(0)487 739 y Fs(U)544 751
y Fo(T)p 644 769 V 700 739 a Fs(g)740 751 y Fn(0)776
739 y Fs(U)833 751 y Fo(S)p 934 769 V 950 769 V 1021
739 a Fu(\()p Fs(S)5 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r
Fu(\))p 1440 769 V 327 w(0)p 1994 769 V 305 w(\()p Fs(S)5
b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 2445 769
V 2462 769 V 304 w(2\()p Fs(S)5 b Fu(\))19 b Fp(\000)f
Fu(2\()p Fs(O)r Fu(\))p 3370 769 V 360 772 3012 4 v 358
872 4 100 v 410 842 a Fs(g)450 854 y Fn(0)487 842 y Fs(U)544
854 y Fo(T)p 644 872 V 697 842 a Fs(g)737 854 y Fn(0)774
842 y Fs(U)831 854 y Fo(T)p 934 872 V 950 872 V 1018
842 a Fu(\()p Fs(T)12 b Fu(\))18 b Fp(\000)h Fu(\()p
Fs(O)r Fu(\))p 1440 872 V 132 w Fp(\000)p Fu(\()p Fs(P)12
b Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p 1994 872 V 115 w(\()p
Fs(S)5 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p
2445 872 V 2462 872 V 117 w(\()p Fs(T)12 b Fu(\))18 b
Fp(\000)h Fu(\()p Fs(P)12 b Fu(\))18 b(+)g(\()p Fs(S)5
b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 3370 872
V 360 875 3012 4 v 358 975 4 100 v 410 945 a Fs(g)450
957 y Fn(0)487 945 y Fs(U)544 957 y Fo(T)p 644 975 V
696 945 a Fs(g)736 957 y Fn(0)773 945 y Fs(U)830 957
y Fo(P)p 934 975 V 950 975 V 1016 945 a Fu(\()p Fs(P)12
b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 1440 975
V 132 w Fp(\000)p Fu(\()p Fs(T)12 b Fu(\))18 b(+)g(\()p
Fs(O)r Fu(\))p 1994 975 V 117 w(\()p Fs(S)5 b Fu(\))19
b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 2445 975 V 2462 975
V 117 w(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(T)12
b Fu(\))18 b(+)g(\()p Fs(S)5 b Fu(\))19 b Fp(\000)f Fu(\()p
Fs(O)r Fu(\))p 3370 975 V 360 978 3012 4 v 354 1084 3025
4 v 352 1184 4 100 v 460 1154 a Fs(g)500 1166 y Fn(1)p
641 1184 V 749 1154 a Fs(g)789 1166 y Fn(2)p 930 1184
V 947 1184 V 999 1154 a Fs(g)1039 1166 y Fn(1)1075 1154
y Fs(\036)1124 1166 y Fo(O)r(; )1247 1154 y Fu(\()p Fs(g)1319
1166 y Fn(2)1356 1154 y Fu(\))p 1437 1184 V 100 w Fp(\000)p
Fs(\036)1602 1166 y Fo(O)r(; )1724 1154 y Fu(\()p Fs(g)1796
1166 y Fn(1)1833 1154 y Fs(g)1873 1166 y Fn(2)1910 1154
y Fu(\))p 1991 1184 V 124 w Fs(\036)2115 1166 y Fo(O)r(; )2238
1154 y Fu(\()p Fs(g)2310 1166 y Fn(1)2347 1154 y Fu(\))p
2451 1184 V 2468 1184 V 471 w(sum)p 3377 1184 V 354 1187
3025 4 v 354 1204 V 352 1303 4 100 v 404 1273 a Fs(g)444
1285 y Fn(0)481 1273 y Fs(U)538 1285 y Fo(P)p 641 1303
V 731 1273 a Fs(U)788 1285 y Fo(O)p 930 1303 V 947 1303
V 1173 1273 a Fu(0)p 1437 1303 V 288 w Fp(\000)p Fu(\()p
Fs(P)12 b Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p 1991 1303
V 115 w(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r
Fu(\))p 2451 1303 V 2468 1303 V 501 w(0)p 3377 1303 V
354 1306 3025 4 v 352 1406 4 100 v 404 1376 a Fs(g)444
1388 y Fn(0)481 1376 y Fs(U)538 1388 y Fo(P)p 641 1406
V 735 1376 a Fs(U)792 1388 y Fo(S)p 930 1406 V 947 1406
V 1015 1376 a Fu(\()p Fs(T)12 b Fu(\))18 b Fp(\000)g
Fu(\()p Fs(O)r Fu(\))p 1437 1406 V 137 w Fp(\000)p Fu(\()p
Fs(S)5 b Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p 1991 1406
V 120 w(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r
Fu(\))p 2451 1406 V 2468 1406 V 118 w(\()p Fs(T)12 b
Fu(\))18 b Fp(\000)g Fu(\()p Fs(S)5 b Fu(\))18 b(+)g(\()p
Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p
3377 1406 V 354 1409 3025 4 v 352 1509 4 100 v 404 1479
a Fs(g)444 1491 y Fn(0)481 1479 y Fs(U)538 1491 y Fo(P)p
641 1509 V 733 1479 a Fs(U)790 1491 y Fo(T)p 930 1509
V 947 1509 V 1018 1479 a Fu(\()p Fs(S)5 b Fu(\))18 b
Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 1437 1509 V 137 w Fp(\000)p
Fu(\()p Fs(T)12 b Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p
1991 1509 V 117 w(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f
Fu(\()p Fs(O)r Fu(\))p 2451 1509 V 2468 1509 V 118 w(\()p
Fs(S)5 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(T)12 b Fu(\))18
b(+)g(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r
Fu(\))p 3377 1509 V 354 1512 3025 4 v 352 1612 4 100
v 404 1582 a Fs(g)444 1594 y Fn(0)481 1582 y Fs(U)538
1594 y Fo(P)p 641 1612 V 731 1582 a Fs(U)788 1594 y Fo(P)p
930 1612 V 947 1612 V 1013 1582 a Fu(\()p Fs(P)12 b Fu(\))19
b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 1437 1612 V 322 w(0)p
1991 1612 V 305 w(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f
Fu(\()p Fs(O)r Fu(\))p 2451 1612 V 2468 1612 V 300 w(2\()p
Fs(P)12 b Fu(\))18 b Fp(\000)g Fu(2\()p Fs(O)r Fu(\))p
3377 1612 V 354 1615 3025 4 v 354 1632 V 352 1732 4 100
v 404 1702 a Fs(g)444 1714 y Fn(0)481 1702 y Fs(U)538
1714 y Fo(P)p 641 1732 V 692 1702 a Fs(g)732 1714 y Fn(0)769
1702 y Fs(U)826 1714 y Fo(O)p 930 1732 V 947 1732 V 1173
1702 a Fu(0)p 1437 1732 V 288 w Fp(\000)p Fu(\()p Fs(P)12
b Fu(\))18 b(+)g(\()p Fs(O)r Fu(\))p 1991 1732 V 115
w(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r
Fu(\))p 2451 1732 V 2468 1732 V 501 w(0)p 3377 1732 V
354 1735 3025 4 v 352 1835 4 100 v 404 1805 a Fs(g)444
1817 y Fn(0)481 1805 y Fs(U)538 1817 y Fo(P)p 641 1835
V 697 1805 a Fs(g)737 1817 y Fn(0)773 1805 y Fs(U)830
1817 y Fo(S)p 930 1835 V 947 1835 V 1018 1805 a Fu(\()p
Fs(S)5 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p
1437 1835 V 137 w Fp(\000)p Fu(\()p Fs(T)12 b Fu(\))18
b(+)g(\()p Fs(O)r Fu(\))p 1991 1835 V 117 w(\()p Fs(P)12
b Fu(\))19 b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 2451 1835
V 2468 1835 V 118 w(\()p Fs(S)5 b Fu(\))18 b Fp(\000)g
Fu(\()p Fs(T)12 b Fu(\))18 b(+)g(\()p Fs(P)12 b Fu(\))19
b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 3377 1835 V 354 1838
3025 4 v 352 1937 4 100 v 404 1908 a Fs(g)444 1920 y
Fn(0)481 1908 y Fs(U)538 1920 y Fo(P)p 641 1937 V 694
1908 a Fs(g)734 1920 y Fn(0)771 1908 y Fs(U)828 1920
y Fo(T)p 930 1937 V 947 1937 V 1015 1908 a Fu(\()p Fs(T)12
b Fu(\))18 b Fp(\000)g Fu(\()p Fs(O)r Fu(\))p 1437 1937
V 137 w Fp(\000)p Fu(\()p Fs(S)5 b Fu(\))18 b(+)g(\()p
Fs(O)r Fu(\))p 1991 1937 V 120 w(\()p Fs(P)12 b Fu(\))19
b Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 2451 1937 V 2468 1937
V 118 w(\()p Fs(T)12 b Fu(\))18 b Fp(\000)g Fu(\()p Fs(S)5
b Fu(\))18 b(+)g(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f
Fu(\()p Fs(O)r Fu(\))p 3377 1937 V 354 1941 3025 4 v
352 2040 4 100 v 404 2011 a Fs(g)444 2023 y Fn(0)481
2011 y Fs(U)538 2023 y Fo(P)p 641 2040 V 693 2011 a Fs(g)733
2023 y Fn(0)770 2011 y Fs(U)827 2023 y Fo(P)p 930 2040
V 947 2040 V 1013 2011 a Fu(\()p Fs(P)12 b Fu(\))19 b
Fp(\000)f Fu(\()p Fs(O)r Fu(\))p 1437 2040 V 322 w(0)p
1991 2040 V 305 w(\()p Fs(P)12 b Fu(\))19 b Fp(\000)f
Fu(\()p Fs(O)r Fu(\))p 2451 2040 V 2468 2040 V 300 w(2\()p
Fs(P)12 b Fu(\))18 b Fp(\000)g Fu(2\()p Fs(O)r Fu(\))p
3377 2040 V 354 2044 3025 4 v 219 2256 a(No)n(w)33 b(one)h(can)g
(rewrite)e(the)i(results)f(of)g(these)h(calculations)c(in)j(more)g
(compact)f(w)n(a)n(y)-7 b(.)55 b(This)33 b(yields)e(the)94
2356 y(third)c(assertion)e(of)j(the)g(prop)r(osition.)p
3585 2306 57 4 v 3585 2356 4 50 v 3638 2356 V 3585 2359
57 4 v 94 2621 a Fk(5.2)113 b Fj(Dihe)-5 b(dr)g(al)33
b(cr)-5 b(oss)35 b(pr)-5 b(o)g(ducts)94 2774 y Fu(No)n(w)26
b(observ)n(e)f(that)h(the)h(image)c(of)j(the)h(co)r(cycle)d
Fs(@)1714 2752 y Fu(^)1703 2774 y Fs(\036)1752 2786 y
Fo(O)r(; )1901 2774 y Fu(lies)g(in)h(\000)h(\(see)g(Section)f(4.3)h
(for)f(the)i(de\014nition)d(of)j(\000\).)94 2873 y(Hence)h(one)f(can)h
(obtain)e(the)i(co)r(cycle)1090 3025 y(\010)1150 3037
y Fo(O)r(; )1295 3025 y Fu(=)23 b Fs(\015)g Fp(\016)18
b Fs(@)1569 3003 y Fu(^)1558 3025 y Fs(\036)1607 3037
y Fo(O)r(; )1753 3025 y Fu(:)23 b Fs(G)18 b Fp(\002)g
Fs(G)24 b Fp(\000)-15 b(!)24 b Fu(\012)f Fp(\032)2383
3003 y Fu(\026)2381 3025 y Fs(k)s Fu(\()2479 3004 y(\026)2459
3025 y Fs(E)5 b Fu(\))2557 2991 y Fm(\003)2619 3025 y
Fs(:)94 3177 y Fu(According)26 b(Prop)r(osition)e(5.1.1)i(the)i(co)r
(cycle)e(\010)1623 3189 y Fo(O)r(; )1773 3177 y Fu(can)h(b)r(e)h
(calculated)d(b)n(y)j(the)g(form)n(ula)422 3642 y(\010)482
3654 y Fo(O)r(; )604 3642 y Fu(\()p Fs(g)676 3654 y Fn(1)713
3642 y Fs(;)14 b(g)790 3654 y Fn(2)827 3642 y Fu(\))23
b(=)970 3272 y Fi(8)970 3347 y(>)970 3372 y(>)970 3397
y(>)970 3422 y(>)970 3446 y(>)970 3471 y(>)970 3496 y(>)970
3521 y(>)970 3546 y(<)970 3696 y(>)970 3720 y(>)970 3745
y(>)970 3770 y(>)970 3795 y(>)970 3820 y(>)970 3845 y(>)970
3870 y(>)970 3895 y(:)1085 3331 y Fu(1)g Fs(;)60 b Fu(\()p
Fs(g)1305 3343 y Fn(1)1342 3331 y Fs(;)14 b(g)1419 3343
y Fn(2)1456 3331 y Fu(\))23 b Fp(2)g Fu(\()p Fs(V)1669
3343 y Fo(O)1745 3331 y Fp(\002)18 b Fs(G)p Fu(\))h Fp([)g
Fu(\()p Fs(G)g Fp(\002)f Fs(V)2265 3343 y Fo(O)2322 3331
y Fu(\))1085 3431 y Fs(x)h Fp(\000)f Fs(x)1281 3443 y
Fo(S)1353 3431 y Fs(;)60 b Fu(\()p Fs(g)1508 3443 y Fn(1)1545
3431 y Fs(;)14 b(g)1622 3443 y Fn(2)1658 3431 y Fu(\))24
b Fp(2)f Fu(\()p Fs(U)1881 3443 y Fo(S)1948 3431 y Fp(\002)18
b Fs(V)2079 3443 y Fo(S)2127 3431 y Fu(\))h Fp([)g Fu(\()p
Fs(g)2324 3443 y Fn(0)2361 3431 y Fs(U)2418 3443 y Fo(T)2488
3431 y Fp(\002)f Fs(V)2619 3443 y Fo(T)2672 3431 y Fu(\))1085
3530 y Fs(x)h Fp(\000)f Fs(x)1281 3542 y Fo(T)1357 3530
y Fs(;)60 b Fu(\()p Fs(g)1512 3542 y Fn(1)1549 3530 y
Fs(;)14 b(g)1626 3542 y Fn(2)1663 3530 y Fu(\))23 b Fp(2)g
Fu(\()p Fs(U)1885 3542 y Fo(T)1956 3530 y Fp(\002)18
b Fs(V)2087 3542 y Fo(T)2140 3530 y Fu(\))h Fp([)f Fu(\()p
Fs(g)2336 3542 y Fn(0)2374 3530 y Fs(U)2431 3542 y Fo(S)2497
3530 y Fp(\002)g Fs(V)2628 3542 y Fo(S)2676 3530 y Fu(\))1085
3630 y Fs(x)h Fp(\000)f Fs(x)1281 3642 y Fo(P)1360 3630
y Fs(;)60 b Fu(\()p Fs(g)1515 3642 y Fn(1)1552 3630 y
Fs(;)14 b(g)1629 3642 y Fn(2)1666 3630 y Fu(\))23 b Fp(2)h
Fs(V)1848 3642 y Fo(P)1922 3630 y Fp(\002)18 b Fs(V)2053
3642 y Fo(P)1162 3693 y(y)p 1095 3711 169 4 v 1095 3758
a(x)p Fm(\000)p Fo(x)1223 3766 y Fg(S)1297 3730 y Fs(;)60
b Fu(\()p Fs(g)1452 3742 y Fn(1)1489 3730 y Fs(;)14 b(g)1566
3742 y Fn(2)1603 3730 y Fu(\))23 b Fp(2)h Fu(\()p Fs(V)1817
3742 y Fo(T)1888 3730 y Fp(\002)18 b Fs(V)2019 3742 y
Fo(P)2075 3730 y Fu(\))h Fp([)f Fu(\()p Fs(U)2288 3742
y Fo(P)2362 3730 y Fp(\002)g Fs(V)2493 3742 y Fo(T)2546
3730 y Fu(\))h Fp([)g Fu(\()p Fs(g)2743 3742 y Fn(0)2780
3730 y Fs(U)2837 3742 y Fo(P)2910 3730 y Fp(\002)f Fs(V)3041
3742 y Fo(S)3090 3730 y Fu(\))1164 3799 y Fo(y)p 1095
3817 174 4 v 1095 3865 a(x)p Fm(\000)p Fo(x)1223 3873
y Fg(T)1301 3836 y Fs(;)60 b Fu(\()p Fs(g)1456 3848 y
Fn(1)1493 3836 y Fs(;)14 b(g)1570 3848 y Fn(2)1607 3836
y Fu(\))23 b Fp(2)h Fu(\()p Fs(V)1821 3848 y Fo(S)1888
3836 y Fp(\002)18 b Fs(V)2019 3848 y Fo(P)2075 3836 y
Fu(\))h Fp([)f Fu(\()p Fs(U)2288 3848 y Fo(P)2362 3836
y Fp(\002)g Fs(V)2493 3848 y Fo(S)2542 3836 y Fu(\))g
Fp([)h Fu(\()p Fs(g)2738 3848 y Fn(0)2775 3836 y Fs(U)2832
3848 y Fo(P)2906 3836 y Fp(\002)f Fs(V)3037 3848 y Fo(T)3090
3836 y Fu(\))1165 3906 y Fo(y)p 1095 3924 175 4 v 1095
3972 a(x)p Fm(\000)p Fo(x)1223 3980 y Fg(P)1303 3943
y Fs(;)60 b Fu(\()p Fs(g)1458 3955 y Fn(1)1495 3943 y
Fs(;)14 b(g)1572 3955 y Fn(2)1609 3943 y Fu(\))23 b Fp(2)h
Fu(\(\()p Fs(U)1864 3955 y Fo(S)1930 3943 y Fp([)19 b
Fs(g)2044 3955 y Fn(0)2081 3943 y Fs(U)2138 3955 y Fo(S)2186
3943 y Fu(\))g Fp(\002)f Fs(V)2368 3955 y Fo(T)2420 3943
y Fu(\))h Fp([)g Fu(\(\()p Fs(U)2666 3955 y Fo(T)2737
3943 y Fp([)g Fs(g)2851 3955 y Fn(0)2887 3943 y Fs(U)2944
3955 y Fo(T)2996 3943 y Fu(\))g Fp(\002)f Fs(V)3178 3955
y Fo(S)3227 3943 y Fu(\))94 4103 y Fv(De\014nition)31
b(5.2.1)40 b Ft(Denote)28 b(by)g Fs(B)1231 4115 y Fo(U)1314
4103 y Ft(the)g(cr)l(oss)g(pr)l(o)l(duct)g(de\014ne)l(d)g(by)g(the)g(c)
l(o)l(cycle)h Fu(\010)2805 4115 y Fo(O)r(; )2955 4103
y Ft(\(c)l(onstructe)l(d)e(by)i(the)94 4203 y(c)l(o)l(cycle)f
Fs(\036)418 4215 y Fo(O)r(; )567 4203 y Ft(of)f(typ)l(e)f
Fs(I)7 b(I)g(I)g Ft(\).)38 b(Then)27 b Fs(B)1333 4215
y Fo(U)1415 4203 y Ft(is)f(a)h(c)l(entr)l(al)f(simple)h(algebr)l(a)h
(with)f(c)l(enter)f Fs(k)s Fu(\()p Fs(E)5 b Fu(\))p Ft(.)38
b(We)26 b(shal)t(l)i(c)l(al)t(l)f Fs(B)3582 4215 y Fo(U)94
4302 y Ft(a)34 b(dihe)l(dr)l(al)i(cr)l(oss)e(pr)l(o)l(duct)g(b)l(e)l(c)
l(ause)f(it)h(c)l(an)f(b)l(e)h(splite)l(d)h(by)f(the)g(normal)g
(extension)2816 4280 y Fu(\026)2814 4302 y Fs(k)2860
4272 y Fo(U)2905 4280 y Fg(O)2992 4302 y Ft(with)g(Galois)h(gr)l(oup)94
4402 y Fs(G=U)258 4414 y Fo(O)337 4380 y Fp(\030)337
4406 y Fu(=)425 4402 y Fs(D)494 4414 y Fn(4)531 4402
y Ft(.)94 4554 y Fv(Prop)s(osition)c(5.2.1)39 b Ft(The)f(dihe)l(dr)l
(al)h(cr)l(oss)d(pr)l(o)l(duct)h Fs(B)1922 4566 y Fo(U)2014
4554 y Ft(is)g(an)g(unr)l(ami\014e)l(d)f(c)l(entr)l(al)g(simple)i
(algebr)l(a)g(over)94 4653 y Fs(k)s Fu(\()p Fs(E)5 b
Fu(\))26 b Ft(of)g(exp)l(onent)f Fu(2)f Ft(and)i(index)g(at)f(most)g
Fu(4)p Ft(.)36 b(The)27 b(class)f(of)g Fs(B)2083 4665
y Fo(U)2163 4653 y Ft(in)g Fu(Br)o(\()p Fs(E)5 b Fu(\))26
b Ft(maps)g(under)f(the)g(homomorphism)94 4753 y Fs(\031)33
b Ft(into)d(the)g(c)l(ohomolo)l(gy)i(class)f(of)f(the)g(c)l(o)l(cycle)h
Fs(\036)1685 4765 y Fo(O)r(; )1808 4753 y Ft(.)94 4905
y(Pr)l(o)l(of)p Fu(.)39 b(T)-7 b(aking)25 b(in)n(to)i(accoun)n(t)f(the)
i(considerations)c(in)j(Section)g(2.3)f(one)i(can)f(calculate)1181
5051 y(in)n(v)o(\(cls)o(\(\010)1509 5063 y Fo(O)r(; )1632
5051 y Fu(\)\))83 b(=)g Fs(\014)t Fu(\()p Fs(\013)p Fu(\(cls\(\010)2281
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5163 y Fo(O)r(; )2487 5151 y Fu(\)\))1779 5260 y(=)83
b Fs(\014)t Fu(\(cls)o(\()p Fs(@)2195 5238 y Fu(^)2184
5260 y Fs(\036)2233 5272 y Fo(O)r(; )2356 5260 y Fu(\)\))1779
5360 y(=)g Fs(\014)t Fu(\()p Fs(\016)s Fu(\(cls\()p Fs(\036)2257
5372 y Fo(O)r(; )2380 5360 y Fu(\)\))1779 5460 y(=)g(0)23
b Fs(:)1825 5697 y Fu(20)p eop
%%Page: 21 21
21 20 bop 94 98 a Fu(Hence)28 b Fs(B)404 110 y Fo(U)488
98 y Fu(is)e(unrami\014ed)f(and)j Fs(\031)s Fu(\([)p
Fs(B)1316 110 y Fo(U)1372 98 y Fu(]\))23 b(=)g(cls)o(\()p
Fs(\036)1712 110 y Fo(O)r(; )1835 98 y Fu(\).)219 214
y(Denote)31 b(b)n(y)f Fs(\032)h Fu(the)g(homomorphism)26
b Fs(H)1511 184 y Fn(2)1548 214 y Fu(\()p Fs(G;)14 b
Fu(\000\))29 b Fp(!)f Fs(H)1982 184 y Fn(2)2019 214 y
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193 y(\026)2231 214 y Fs(E)5 b Fu(\))2329 184 y Fm(\003)2368
214 y Fu(\))31 b(induced)f(b)n(y)g(the)i(comp)r(osition)27
b(\000)3577 163 y Fo(\015)3555 214 y Fp(!)94 314 y Fu(\012)d
Fp(\032)267 292 y Fu(\026)265 314 y Fs(k)s Fu(\()362
293 y(\026)343 314 y Fs(E)5 b Fu(\))441 284 y Fm(\003)480
314 y Fu(.)37 b(Then)1274 422 y(2)14 b(cls)n(\(\010)1514
434 y Fo(O)r(; )1636 422 y Fu(\))84 b(=)e(2)14 b(cls)n(\()p
Fs(\015)24 b Fp(\016)18 b Fs(@)2265 401 y Fu(^)2255 422
y Fs(\036)2304 434 y Fo(O)r(; )2426 422 y Fu(\))1752
532 y(=)82 b(2)p Fs(\032)14 b Fu(cls)n(\()p Fs(@)2182
510 y Fu(^)2171 532 y Fs(\036)2220 544 y Fo(O)r(; )2343
532 y Fu(\))1752 632 y(=)82 b(2)p Fs(\032\016)s Fu(\(cls)o(\()p
Fs(\036)2230 644 y Fo(O)r(; )2353 632 y Fu(\)\))1752
731 y(=)g Fs(\032\016)s Fu(\(cls)o(\(2)p Fs(\036)2230
743 y Fo(O)r(; )2353 731 y Fu(\)\))1752 831 y(=)g(0)23
b Fs(:)94 976 y Fu(Hence)28 b Fs(B)404 988 y Fo(U)488
976 y Fu(has)f(exp)r(onen)n(t)g(2.)219 1076 y(Since)g(the)g
(restriction)e(\010)1036 1088 y Fo(O)r(; )1153 1076 y
Fp(j)1176 1088 y Fo(V)1215 1096 y Fg(O)1265 1088 y Fm(\002)p
Fo(V)1356 1096 y Fg(O)1437 1076 y Fu(is)h(zero)g(\(see)i(explicit)c
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y Fu(b)r(elo)n(w\),)g(the)h(algebra)c Fs(B)3582 1088
y Fo(U)94 1176 y Fu(splits)g(b)n(y)h(the)g(\014eld)743
1154 y(\026)741 1176 y Fs(k)787 1146 y Fo(V)826 1154
y Fg(O)879 1176 y Fu(.)36 b(But)25 b(the)h(subgroup)e
Fs(V)1647 1188 y Fo(O)1729 1176 y Fu(has)h(index)f(4)h(in)f
Fs(G)h Fu(b)n(y)g(Prop)r(osition)d(5.1.1.)35 b(Consequen)n(tly)-7
b(,)94 1275 y(the)28 b(index)f(of)h Fs(B)619 1287 y Fo(U)702
1275 y Fu(is)e(at)i(most)e(4.)p 3585 1226 57 4 v 3585
1275 4 50 v 3638 1275 V 3585 1278 57 4 v 94 1558 a Fk(5.3)113
b Fj(End)34 b(of)h(the)g(pr)-5 b(o)g(of)94 1711 y Fu(Let)33
b Fs(c)e Fp(2)416 1723 y Fn(2)453 1711 y Fs(H)529 1681
y Fn(1)566 1711 y Fu(\()p Fs(G;)720 1690 y Fu(\026)700
1711 y Fs(E)6 b Fu(\).)52 b(According)30 b(to)j(Prop)r(osition)c(3.3.1)
i(the)i(class)d Fs(c)j Fu(can)f(b)r(e)h(presen)n(ted)f(b)n(y)g(a)g(co)r
(cycle)f Fs(\036)94 1810 y Fu(of)k(one)f(of)g(three)g(t)n(yp)r(es)g
Fs(I)7 b Fu(,)36 b Fs(I)7 b(I)42 b Fu(or)33 b Fs(I)7
b(I)g(I)g Fu(.)57 b(Let)34 b Fs(U)43 b Fu(=)34 b(k)n(er)o(\()p
Fs( )s Fu(\),)j(where)c Fs( )38 b Fu(=)33 b Fs(\036)7
b Fp(j)2602 1822 y Fo(H)2665 1810 y Fu(.)57 b(If)35 b
Fs(\036)g Fu(is)e(a)h(co)r(cycle)e(of)j(t)n(yp)r(e)94
1910 y Fs(I)7 b Fu(,)41 b(then)d Fs(\031)s Fu(\([\()p
Fs(a;)14 b(x)27 b Fp(\000)d Fs(x)828 1922 y Fo(P)884
1910 y Fu(\)]\))41 b(=)e Fs(c)p Fu(,)i(where)c Fs(a)j
Fp(2)g Fs(k)h Fu(suc)n(h)c(that)2118 1888 y(\026)2116
1910 y Fs(k)2162 1880 y Fo(U)2257 1910 y Fu(=)j Fs(k)s
Fu(\()2440 1850 y Fp(p)p 2509 1850 44 4 v 60 x Fs(a)p
Fu(\))e(\(see)g(Prop)r(osition)c(4.1.1\).)66 b(So)94
2010 y(one)36 b(can)f(put)i Fs(b)612 2022 y Fo(c)682
2010 y Fu(=)g([\()p Fs(a;)14 b(x)24 b Fp(\000)g Fs(x)1127
2022 y Fo(P)1183 2010 y Fu(\)])37 b Fp(2)1381 2022 y
Fn(2)1418 2010 y Fs(\031)1468 1980 y Fm(\000)p Fn(1)1558
2010 y Fu(\()p Fs(c)p Fu(\))f(\(see)g(Section)f(2.5\).)61
b(If)36 b Fs(\036)h Fu(is)e(a)g(co)r(cycle)f(of)i(t)n(yp)r(e)g
Fs(I)7 b(I)g Fu(,)38 b(then)94 2109 y Fs(\031)s Fu(\([\()p
Fs(Z)q(;)14 b(f)9 b Fu(\)]\))31 b(=)e Fs(c)p Fu(,)k(where)e
Fs(Z)k Fu(=)1112 2087 y(\026)1110 2109 y Fs(k)1156 2079
y Fo(U)1243 2109 y Fu(\(see)c(Prop)r(ositions)d(4.3.1)i(and)h(4.4.1\))g
(suc)n(h)g(that)2876 2087 y(\026)2874 2109 y Fs(k)2920
2079 y Fo(U)3005 2109 y Fu(=)e Fs(k)s Fu(\()3177 2049
y Fp(p)p 3246 2049 V 60 x Fs(a)p Fu(\).)49 b(So)31 b(one)94
2209 y(can)23 b(tak)n(e)f Fs(b)453 2221 y Fo(c)510 2209
y Fu(=)g([\()p Fs(a;)14 b(x)9 b Fp(\000)g Fs(x)910 2221
y Fo(P)967 2209 y Fu(\)].)35 b(A)n(t)24 b(last,)e(if)h
Fs(\036)g Fu(is)f(a)h(co)r(cycle)e(of)i(t)n(yp)r(e)g
Fs(I)7 b(I)g(I)g Fu(,)24 b(then)g(let)e Fs(b)2725 2221
y Fo(c)2781 2209 y Fu(=)h([)p Fs(B)2955 2221 y Fo(U)3011
2209 y Fu(])g(\(see)g(Prop)r(osition)94 2309 y(5.2.1\).)36
b(No)n(w)27 b(applying)f(Lemma)f(2.4.1)h(w)n(e)h(\014nish)g(the)h(pro)r
(of)f(of)h(Theorem)e(1.0.1.)94 2583 y Fw(References)94
2765 y Fu([Lic)n(h69])175 b(S.Lic)n(h)n(ten)n(baum.)34
b Ft(Duality)k(The)l(or)l(ems)i(for)f(Curves)f(over)h
Fs(P)12 b Ft(-adic)39 b(Fields.)g Fu(In)n(v)n(en)n(t.)d(Math.)h
Fv(7)p Fu(,)556 2865 y(120)26 b(-)h(136)f(\(1969\).)94
3031 y([Co88])230 b(J.-L.Colliot-Th)n(\023)-39 b(el)m(\022)f(ene)25
b(\(with)30 b(the)h(collab)r(oration)26 b(of)31 b(J.-J.Sansuc\).)e
Ft(The)34 b(r)l(ationality)g(pr)l(oblem)556 3130 y(for)27
b(\014elds)g(of)g(invariants)g(under)f(line)l(ar)h(algebr)l(aic)i(gr)l
(oups)d(\(with)h(sp)l(e)l(cial)h(r)l(e)l(gar)l(d)f(to)f(the)g(Br)l
(auer)556 3230 y(gr)l(oup\).)i Fu(Unpublished)f(Lecture)g(Notes)g(from)
f(the)i(9th)g(ELAM,)g(San)n(tiago)d(de)i(Chile,)f(1988.)94
3396 y([GMY97])129 b(V.I.Guletski)-9 b(\024)-32 b(\020,)49
b(G.L.Margolin,)h(V.I.Y)-7 b(anc)n(hevski)e(\024)-32
b(\020.)44 b Ft(Pr)l(esentation)k(of)i(two-torsion)e(p)l(art)h(in)556
3496 y(Br)l(auer)25 b(gr)l(oups)g(of)g(curves)g(by)g(quaternion)g
(algebr)l(as.)f Fu(\(In)e(Russian\).)f(Dokl.)g(NANB.)i(41)e(\(1997\),)
556 3595 y(no.)27 b(6,)g(4)h(-)f(8.)94 3761 y([GY98])205
b(V.I.Guletski)-9 b(\024)-32 b(\020,)25 b(V.I.Y)-7 b(anc)n(hevski)e
(\024)-32 b(\020.)25 b Ft(Pr)l(esentation)30 b(of)i(torsion)f(in)f(Br)l
(auer)h(gr)l(oups)g(of)g(curves)f(by)556 3861 y(cyclic)h(algebr)l(as.)p
Fu(\(In)f(Russian\).)c(Dokl.)h(AN)h(Belarusi,)d(42)h(\(1998\),)h(no.)g
(2,)g(52-55.)94 4027 y([YM96])194 b(V.I.Y)-7 b(anc)n(hevski)e(\024)-32
b(\020,)38 b(G.L.Margolin.)d Ft(The)41 b(Br)l(auer)f(gr)l(oups)g(and)g
(the)g(torsion)g(of)h(lo)l(c)l(al)g(el)t(liptic)556 4126
y(curves.)28 b Fu(St.P)n(eterburgsburg)d(Math.)j(J.)f(V)-7
b(ol.7)27 b(\(1996\),)f(no.3.)94 4293 y([VY96])208 b(G.B.V)-7
b(asiuk,)34 b(V.I.Y)-7 b(anc)n(hevski)e(\024)-32 b(\020.)30
b Ft(On)k(the)i(2-torsion)g(p)l(art)f(of)h(Br)l(auer)g(gr)l(oup)f(of)i
(lo)l(c)l(al)f(el)t(liptic)556 4392 y(curves)25 b(with)h(go)l(o)l(d)g
(r)l(e)l(duction)f(de\014ne)l(d)h(over)g(the)f(\014eld)h(of)g(2-adic)h
(numb)l(ers.)p Fu(\(In)22 b(Russian\).)g(Dokl.)556 4492
y(AN)28 b(Belarusi,)c(40)j(\(1996\),)f(no.6,)h(31-33.)94
4658 y([V97])270 b(G.B.V)-7 b(asiuk.)36 b Ft(On)i(the)h(2-torsion)g(p)l
(art)g(of)h(Br)l(auer)f(gr)l(oup)g(of)h(2-adic)g(el)t(liptic)h(curves)e
(in)g(the)556 4757 y(nonsplit)26 b(c)l(ase)g(of)g(b)l(ad)g(r)l(e)l
(duction.)p Fu(\(In)e(Russian\).)e(Dokl.)g(AN)i(Belarusi,)d(41)h
(\(1997\),)h(no.6,)g(13-16.)427 4924 y([1])41 b(M.)23
b(v)-5 b(an)22 b(den)h(Bergh)f Ft(The)k(algebr)l(aic)h(index)f(of)g(a)g
(division)h(algebr)l(a.)e Fu(Lect.)d(Notes)h(in)f(Math.)h(1197)556
5023 y(\(1985\))j(190-206.)94 5189 y([F)-7 b(add51])151
b(D.K.F)-7 b(addeev.)22 b Ft(Simple)k(algebr)l(as)g(over)g(a)f
(function)g(\014eld)g(in)g(one)h(variable.)f Fu(Pro)r(c.)c(Steklo)n(v)g
(Inst.)556 5289 y(38)26 b(\(1951\))h(321)f(-)h(344;)g(English)d
(transl.)i(in)h(AMS)h(T)-7 b(ransl.)26 b(3)h(\(1956\))g(15)f(-)i(38.)
1825 5697 y(21)p eop
%%Page: 22 22
22 21 bop 94 98 a Fu([Mi81])232 b(J.S.Milne.)38 b Ft(Comp)l(arison)k
(of)g(the)f(Br)l(auer)g(gr)l(oup)g(with)h(the)f(T)-6
b(ate-)2759 77 y(\024)2752 98 y(Safar)l(evi)n(\024)-40
b(c)42 b(gr)l(oup.)f Fu(J.)e(F)-7 b(ac.)556 197 y(Science,)26
b(Univ.)h(T)-7 b(oky)n(o,)26 b(Sec.)i(IA.)g(28)f(\(1981\))f(735-743)f
(.)94 364 y([Sc)n(h69])205 b(W.Sc)n(harlau.)1039 343
y Ft(\177)1024 364 y(Ub)l(er)27 b(bie)i(Br)l(auer-Grupp)l(e)f(eines)g
(algebr)l(aischen)h(F)-6 b(unktionen-k\177)-42 b(orp)l(ers)28
b(in)f(einer)556 463 y(V)-6 b(ariab)l(ein.)29 b Fu(J.)e(f)r(\177)-44
b(ur)28 b(die)f(reine)f(und)i(angew.)f(Math.)g(Bd.)h(239/240)c
(\(1969\),)j(S.)h(1-6.)94 629 y([Serre])230 b(J.-P)-7
b(.Serre.)25 b Ft(Cohomolo)l(gie)33 b(Galoisienne.)d
Fu(Springer-V)-7 b(erlag,)24 b(1964.)94 795 y([Hall59])180
b(M.Hall.)26 b Ft(The)k(the)l(ory)h(of)f(gr)l(oups.)f
Fu(N.Y.)f(MacMillan,)c(1959.)94 961 y([Mac63])177 b(S.Maclane.)22
b Ft(Homolo)l(gy.)j Fu(Die)e(grundlehren)f(der)h(mathematisc)n(hen)d
(wissensc)n(haften.)i(Springer-)556 1061 y(V)-7 b(erlag,)25
b(1963.)94 1227 y([Pie82])214 b(R.S.Pierce.)22 b Ft(Asso)l(ciative)28
b(algebr)l(as.)e Fu(Graduate)e(T)-7 b(exts)24 b(in)f(Mathematics)f(88,)
i(Springer-V)-7 b(erlag,)556 1327 y(1982.)1825 5697 y(22)p
eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF

--Boundary_(ID_25zsK+32yMjxPHWmyyrWXQ)--

