\part{Chapter IV}{Specialization with Respect to Quadratic Places}
\leftrunningtitle{Chapter IV. Specialization with Respect to Quadratic Places}

\titlea{\S28}{Quadratic Places; Specialization of Bilinear Forms}
\leftrunningtitle{Chapter IV. Specialization with Respect to Quadratic Places}
%\leftrunningtitle{Specialization of Quadratic and Symmetric
%Bilinear Forms}
\rightrunningtitle{\S28 Quadratic Places; Specialization of Bilinear Forms}

In this section we mean by a ``form'' always a nondegenerate symmetric bilinear form.

To begin with we recall a result from \S3. Let $\lambda
\colon K \to L \cup \infty$ be a place and ${\nam o} = {\nam
o}_{\lambda}$ the associated valuation ring. We defined an additive map
$$\lambda_W \colon  W(K) \longrightarrow W(L),$$
which has the following values for one-dimensional forms ($=$ square classes):
\medskip	
\setitemindent{(2)}

\item{(1)} $\lambda_W \langle \varepsilon \rangle = \langle \lambda
(\varepsilon) \rangle$ for $\varepsilon \in {\nam o}^*$,
\smallskip	
\item{(2)} $\lambda_W (\alpha) = 0$ for $\alpha \in Q(K) \setminus Q
({\nam o})$.
\medskip	

\noindent Here we identified the square class group  $Q({\nam o}) = {\nam
o}^*/{\nam o}^{*2}$ with its image in $Q(K)$ under the natural map
$\varepsilon {\nam o}^{*2} \mapsto \varepsilon K^{*2}$.

More generally we have that the natural map $\{ M \} \mapsto \{ K
\otimes_{{\nam o}} M \}$ from $W({\nam o})$ to $W(K)$ is injective. 
(Here $\{ M \}$ denotes the Witt class of a nondegenerate
 ${\nam o}$-module $M$). Thus we can also regard
$W({\nam o})$ as a subring of
$W(K)$. The restriction of  $\lambda_W$ to
$W({\nam o})$ can then be interpreted as the ring homomorphism $W({\nam o}) \to W(L)$, induced by the ring homomorphism  $\lambda \vert {\nam o} \colon {\nam o} \to L$.
The restriction  $\lambda_W \vert W({\nam o})$ is thus also multiplicative. Furthermore,
 $\lambda_W \colon W(K) \to W(L)$ is
semi-linear with respect to $W({\nam o}) \to W(L)$. In other words, we have for 
$\xi \in W({\nam o})$ and $\eta \in W(K)$, that
$$\lambda_W (\xi \eta)  =  \lambda_W (\xi) \cdot \lambda_W (\eta).$$
Starting from the map  $\lambda_W$ we developed a specialization theory for bilinear forms in \S3.
In particular  we associated to a form  $\varphi$ with good reduction with respect to $\lambda$ 
(i.e., a form over $K$ that comes from a nondegenerate bilinear module over ${\nam o}$)
a form  $\lambda_* (\varphi)$ over $L$ such that $\{
\lambda_* (\varphi) \} = \lambda_W (\{ \varphi \})$ and $\dim
\lambda_* (\varphi) = \dim \varphi$. 
The form $\lambda_* (\varphi)$ is only determined up to stable isometry by the form
$\varphi$ however.  In truth we thus associated to $\varphi$ 
only a stable isometry class of forms over $L$. Nonetheless we will disregard this fact and just speak of $\lambda_*(\varphi)$ as if it were a form over $L$.

Note that if $\Char L \ne 2$, then stable isometry is the same  as  isometry by Witt's Cancellation Theorem and so  $\lambda_* (\varphi)$ is uniquely determined --
up to isomorphism -- by $\lambda$ and $\varphi$.

Now we would like to associate to a ``quadratic place'' $\Lambda \colon K
\to L \cup \infty$ (see the following definition) a map
$\Lambda_W \colon W(K) \to W(L)$ in a similar manner as was done above
and, using this foundation, develop a specialization theory for quadratic places that generalizes the theory developed in \S3.

\begdefinition{1.} Let $K$ and $L$ be fields. A  {\it
quadratic place} ($= Q$-place)\Idx{quadratic place}\Idx{Q-place@$Q$-place} 
from $K$ to $L$ is a triple 
$(\lambda, H, \chi)$ consisting of a place $\lambda \colon K
\to L \cup \infty$ (in the usual sense), a subgroup $H$ of
$Q(K)$  that contains $Q({\nam o})$  (with ${\nam o} \colon = {\nam
o}_{\lambda}$) and a group homomorphism  $\chi \colon H \to
Q(L)$ with $\chi (\langle \varepsilon \rangle) = \langle \lambda
(\varepsilon) \rangle$ for every $\varepsilon \in {\nam o}^*$,
and thus $\chi (\alpha) = \lambda_* (\alpha) = \lambda_W (\alpha)$ for
every $\alpha \in Q ({\nam o})$. 
\enddefinition

We will often denote the triple  $(\lambda, H, \chi)$ by a Greek capital letter, for instance 
$\Lambda$, and use the symbolic notation
$\Lambda \colon K \to L \cup \infty$. We call a group homomorphism
from a subgroup $H$ of $Q(K)$ to 
$Q(L)$ a  ($Q(L)$-valued) {\it character} of $H$.\Idx{QL-valued character@ ($Q(L)$-valued) character}

\begexample{} To every place  $\lambda \colon K \to L \cup \infty$
we can associate a quadratic place  $\widehat{\lambda} \colon =
(\lambda, H_{\lambda}, \chi_{\lambda})$\Idx{111lhat@$\widehat{\lambda}$}
by letting
$H_{\lambda} = Q({\nam o})$ and taking for  $\chi_{\lambda}$ 
the character $\langle \varepsilon \rangle
\mapsto \langle \lambda (\varepsilon) \rangle$ given by
$\lambda$.
This $Q$-place $\widehat{\lambda}$ can then be ``extended'' to a $Q$-place
$\Lambda \colon = (\lambda, H, \chi)$ if we choose a subgroup
$H$ of $Q(K)$ which contains $Q({\nam o})$ and somehow continue the
character $\chi_{\lambda}$ to a character
$\chi \colon H \to Q(L)$. Such a continuation is always possible since $H$ and $Q(L)$ are elementary abelian of exponent $2$, and can thus be interpreted as 
${\Bbb F}_2$-vector spaces.
\endexample

In this set-up we imagine that  $\widehat{\lambda}$ is the same object as 
$\lambda$, but just dressed up, so that the notion of quadratic place generalizes the notion of place.

\begtheorem{1.} Let $\Lambda = (\lambda, H, \chi)$ be a   $Q$-place from
$K$ to $L$.
\smallskip	
\setitemindent{$(b)$}

\item{$(a)$}  There is precisely one additive map
$$\Lambda_W \colon \quad W(K) \longrightarrow W(L)$$
with the following properties:\Idx{111lw4@$\Lambda_W$}
\smallskip	
\setitemitemindent{$(4)$}
\itemitem{$(3)$}  $\Lambda_W (\alpha) = \chi (\alpha)$, if $\alpha \in H$.
\smallskip	
\itemitem{$(4)$}  $\Lambda_W (\alpha) = 0$, if $\alpha \in Q(K) \setminus
H$.
\medskip	

\item{$(b)$} The restriction of $\Lambda_W$ to the ring $W(K, H)$,\Idx{111wkh@$W(K, H)$} 
generated by
$H$ in $W(K)$, is a ring homomorphism from $W(K,
H)$ to $W(L)$, and $\Lambda_W$ is semilinear with respect to this ring homomorphism.
\endtheorem

\begproof{} The uniqueness  of  $\Lambda_W$ is clear since 
$W(K)$ is additively generated by  $Q(K)$. In order to prove the existence, we choose a subgroup  $H_0$ of $H$ with $H = H_0 \times Q
({\nam o})$ (i.e., a ``complement'' $H_0$ of $Q({\nam o})$ in
$H$). Then we define
$$\Lambda_W (\xi) \colon  = \sum\limits_{\alpha \in H_0} \chi (\alpha)
\lambda_W (\alpha \xi) \qquad (\xi \in W(K)). \leqno(5)$$
This map $\Lambda_W \colon Q(K) \to Q(L)$ is clearly additive. It satisfies  conditions
 (3) and (4), and also possesses the properties stated in part $(b)$, as can be verified instantly.\qed
\endproof 

Assume now that  $\Lambda = (\lambda, H, \chi) \colon K \to L
\cup \infty$ is a $Q$-place and that ${\nam o} \colon = {\nam o}_{\lambda}$. We 
choose a subgroup $H_0$ of $H$ with $H \colon = H_0 \times Q
({\nam o})$. Let $\varphi$ be a form over $K$.

\begdefinition{2.} We say that  $\varphi$ has {\it good reduction} (abbreviated
GR) {\it with respect to}\Idx{good reduction (w.r.t. a Q-place)@good reduction (w.r.t. a $Q$-place)}\Idx{111gr2@GR (w.r.t. a $Q$-place)} 
$\Lambda$, when $\varphi$ has a decomposition (modulo
stable isometry)
$$\varphi  \approx  \raise5pt\hbox{$\newbot\limits_{\alpha \in H_0}$} \alpha \varphi_{\alpha},
\leqno(6)$$
in which the forms  $\varphi_{\alpha}$ are $\lambda$-unimodular.
We then also say that $\varphi$ is $\Lambda$-{\it unimodular}.\Idx{l2-unimodular@$\Lambda$-unimodular} 
Here we used the notation $\alpha \varphi_{\alpha} \colon = \alpha \otimes
\varphi_{\alpha}$. (If $\alpha = \langle a \rangle$, then $\alpha
\varphi_{\alpha} = a \varphi_{\alpha}$ up to isometry.)

One should note that  $\Lambda$-unimodularity is independent of the choice of the complement  $H_0$ of $Q({\nam o})$ in $H$.
\enddefinition

\begexample{} If  $H = Q(K)$, then {\it every} form $\varphi$
over $K$ has good reduction with respect to $\Lambda$. Namely, if $\varphi
\cong r \times \left( {0 \atop 1} {1 \atop 0} \right)$ is hyperbolic, then 
 $\varphi$ is actually $\lambda$-unimodular. Otherwise we choose a diagonalization
$\varphi \cong \langle a_1, \dots, a_n
\rangle$ and take  those elements $\langle a_j \rangle \in Q(K)$ together that are in the same
coset of $Q({\nam o})$.
\endexample

\begdefinition{3.} Assume that  $\varphi$ has GR with respect to $\Lambda$. We choose 
a decomposition  (6) of $\Lambda$ and let
$$\Lambda_* (\varphi) \colon  =  \raise5pt\hbox{$\newbot\limits_{\alpha \in
H_0}$} \chi (\alpha) \lambda_* (\varphi_{\alpha}). \leqno(7)$$
We call $\Lambda_* (\varphi)$ a {\it specialization of}
$\varphi$ {\it with respect to} $\Lambda$.\Idx{specialization (w.r.t. a Q-place)@specialization (w.r.t. a $Q$-place)}\Idx{111lsfi3@$\Lambda_* (\varphi)$}
\enddefinition

\begtheorem{2.} If  $\varphi$ has GR with respect to $\Lambda$ then, up to stable isometry, the form
$\Lambda_* (\varphi)$ does not depend on the choice of the decomposition $(6)$, and also not on the choice of the complement
$H_0$ of $Q({\nam o})$ in $H$.
\endtheorem

\begproof{} We clearly have
$$\Lambda_W (\{ \varphi \})  =  \{ \Lambda_* (\varphi) \}. \leqno(8)$$
Moreover,  $\dim \Lambda_* (\varphi) = \dim \varphi$. Noting that a form over
$L$ is uniquely determined up to stable isometry by its Witt class and dimension finishes the proof.\qed
\endproof

In the future we will call  $\Lambda_* (\varphi)$ ``the'' specialization of  $\varphi$
with respect to  $\Lambda$. Although sloppy, we hope this notation is harmless and helpful.  

\begremark{} This notion of specialization generalizes the one from \S3: if 
$\lambda$ is a place from $K$ to $L$, then a form 
$\varphi$ over $K$ has GR with respect to $\widehat{\lambda}$ if and only if
it has GR with respect to $\lambda$, and then we have 
$\widehat{\lambda}_* (\varphi) \approx \lambda_* (\varphi)$. 
\endremark

Our goal is now to use these notions of good reduction and specialization to build a theory, analogous to the theory developed in \S3 and then -- in the case $\Char L\ne 2$ -- to obtain results about the generic splitting of  $\Lambda_* (\varphi)$ 
from the generic splitting behaviour of  $\varphi$, analogous to \S 4, see
\S 30 below.

It is obvious that for forms $\varphi, \psi$ over
$K$ which have   GR with respect to $\Lambda$,   also $\varphi \perp \psi$ and $\varphi\otimes \psi$ have GR with respect to $\Lambda$ and that
$$\Lambda_* (\varphi \perp \psi)  \approx  \Lambda_* (\varphi) \perp
\Lambda_* (\psi), \quad \Lambda_* (\varphi \otimes \psi)  \approx
 \Lambda_* (\varphi) \otimes \Lambda_* (\psi). \leqno(9)$$
It is not clear however that the analogue of  \S 3,
Theorem~4  remains valid. When $\varphi$ and $\varphi \perp \psi$ have GR
with respect to $\Lambda$, why should   $\psi$ then also have  GR with respect to $\Lambda$?

\S3, Theorem~4  was crucial for the development of the generic splitting theory of \S4. In order to save this theorem we will now broaden the notion of ``good reduction'' to the new notion of  ``almost good reduction''.

Given a $Q$-place  $\Lambda = (\lambda, H, \chi)$, we choose 
a subgroup  $S$ of $Q(K)$ with $Q(K) = S \times H$. For every
form $\varphi$ over $K$ we then have a decomposition
$$\varphi  \approx  \raise5pt\hbox{$\newbot\limits_{s \in S}$} s \varphi_s
\leqno(10)$$
with  $\Lambda$-unimodular forms  $S$ (even with $\cong$ instead of
$\approx$). This is obvious for $\varphi \cong \langle a
\rangle$  one-dimensional and for $\varphi \cong \left( {0 \atop 1}
{1 \atop 0} \right)$, and so clear for every form $\varphi$ over
$K$. We call the decomposition (10) a $\Lambda$-{\it modular
decomposition} of $\varphi$.\Idx{l2-modular decomposition@$\Lambda$-modular decomposition}

\beglemma{1.}  For every $s \in S$,  the Witt class of $\Lambda_*
(\varphi_s)$ in a  $\Lambda$-modular decomposition  $(10)$ of
$\varphi$  is uniquely determined by  $\varphi$, $\Lambda$ and $s$.
\endlemma

\begproof{} For every  $s \in S$ we choose a decomposition (see
(6) above)
$$\varphi_s  \approx  \raise5pt\hbox{$\newbot\limits_{\alpha \in H_0}$}
\alpha \varphi_{s, \alpha}$$
with  $\lambda$-unimodular forms $\varphi_{s, \alpha}$. Then
$$\varphi  \approx \raise5pt\hbox{$\newbot\limits_{(s, \alpha) \in S \times
H_0}$} s \alpha  \varphi_{s, \alpha}.$$
Clearly,  $\lambda_W ( \{ s \alpha \varphi \} ) = \{
\lambda_* (\varphi_{s, \alpha}) \}$. Hence the Witt class of
$\lambda_* (\varphi_{s, \alpha})$ is uniquely determined by
$\varphi$ and $\Lambda$ for fixed $s$ and $\alpha$. It follows that the Witt class of
$$\Lambda_* (\varphi_s)  \approx  \raise5pt\hbox{$\newbot\limits_{\alpha \in
H_0}$} \chi (\alpha) \lambda_* (\varphi_{s, \alpha})$$
is uniquely determined by  $\varphi$ and $\Lambda$  for fixed
$s$ and $H_0$. By Theorem~2, $\Lambda_*
(\varphi_s)$ does not change, up to stable isometry, if we pass from
$H_0$ to a different complement of  $Q ({\nam o})$ in $H$.\qed
\endproof

\begdefinition{4.} Consider a  $\Lambda$-modular decomposition
 (11) of $\varphi$. We say that  $\varphi$ has {\it almost
good reduction} (abbreviated AGR)\Idx{almost good reduction (w.r.t. a Q-place)@almost good reduction (w.r.t. a $Q$-place)}\Idx{111agr@AGR} 
{\it with respect to} $\Lambda$ if all forms\Idx{111lsfi3@$\Lambda_* (\varphi)$}
 $\Lambda_* (\varphi_s)$ with $s \in S$, $s \ne 1$, are metabolic. We then define
$$\Lambda_* (\varphi) \colon  =  \Lambda_* (\varphi_1) \perp \frac{\dim \varphi
- \dim \varphi_1}{2} \times \tilde{H},$$
with  $\tilde{H} \colon = \left( {0 \atop 1} {1 \atop 0}
\right)$, and call $\Lambda_* (\varphi)$ ``the'' {\it
specialization of $\varphi$ with respect to $\Lambda$}.\Idx{specialization (w.r.t. a Q-place)@specialization (w.r.t. a $Q$-place)}.\Idx{111hist@$\tilde{H} \colon = \left( {0 \atop 1} {1 \atop 0}
\right)$}
\enddefinition

It is clear from  Lemma~1 that the property AGR is independent of the choice of
 $\Lambda$-modular decomposition (10), and that also the Witt class
of $\Lambda_* (\varphi)$ does not depend on this choice. Since  $\dim \Lambda_* (\varphi) = \dim \varphi$ we have that 
$\Lambda_* (\varphi)$ is uniquely determined by $\varphi$ up to stable isometry.  It is now also clear that the property AGR and the stable isometry class of $\Lambda_* (\varphi)$ are independent of the choice of the complement  $S$ of $H$ in
$Q(K)$.

One should note that for the property AGR all three components
$\lambda, H, \chi$ of $\Lambda = (\lambda, H, \chi)$ play a role, whereas  GR
only depends on $\lambda$ and $H$.

One easily verifies the following

\begtheorem{3.} Let $\varphi$ and $\psi$ be forms over  $K$. If 
$\varphi$ and $\psi$ have almost good reduction with respect to $\Lambda$, this is also the case for
 $\varphi \perp \psi$ and $\varphi \otimes \psi$, and we have
$$\leqalignno{
\Lambda_* (\varphi \perp \psi)  &\approx 
\Lambda_* (\varphi) \perp \Lambda_* (\psi),  &{(11)}\cr
\Lambda_* (\varphi \otimes \psi) 
&\approx  \Lambda_* (\varphi) \otimes \Lambda_* (\psi). & {(12)}}$$
\endtheorem

In addition, the analogue of \S3, Theorem~4 also holds for almost good reduction. The proof of this analogue -- in the current state of affairs -- will turn out to be trivial.

\begtheorem{4.} Let $\varphi$ and $\psi$ be forms over $K$. Assume that the forms
 $\varphi$ and $\varphi \perp \psi$ have almost good reduction with respect to $\Lambda$. Then $\psi$ also has almost good reduction with respect to  $\Lambda$.
\endtheorem 

\begproof{} We choose  $\Lambda$-modular decompositions
$$\varphi  \approx  \raise5pt\hbox{$\newbot\limits_{s \in S}$} s \varphi_s, \quad \psi  \approx
 \raise5pt\hbox{$\newbot\limits_{s \in S}$} s \psi_s.$$
Then
$$\varphi \perp \psi  \approx  \raise5pt\hbox{$\newbot\limits_{s \in S}$}
s (\varphi_s \perp \psi_s)$$
is a $\Lambda$-modular decomposition of $\varphi \perp \psi$. For every
$s \in S$ we have
$$\Lambda_* (\varphi_s \perp \psi_s)  \approx  \Lambda_* (\varphi_s) \perp
\Lambda_* (\psi_s).$$
If $s \ne 1$, then $\Lambda_* (\varphi_s) \sim 0$ and
$\Lambda_* (\varphi_s \perp \psi_s) \sim 0$, and thus also $\Lambda_*
(\psi_s) \sim 0$.\qed
\endproof


