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\begin{center}
\underline{\large \bf Homogeneous varieties - zero cycles of}\\
\underline{\large\bf degree one versus rational points}\\[2mm]
{\bf R. Parimala}
\end{center}

{\bf Abstract}
{\it 
Examples of projective homogeneous varieties over
the field of Laurent series over p-adic fields
which admit zero cycles of degree one and which do
not have rational points are constructed.}

\vspace{1.0cm}

Let $k$ be a field and $X$ a quasi-projective variety over $k$.  Let 
$Z_0(X)$ denote the group of zero cycles on $X$ 
and deg : $Z_0(X) \to \ZZ$ the degree 
homomorphism which associates to a closed point $x$ of $X$, the degree 
$[k(x):k]$ of its residue field. 

We study the question: for what classes $\mathfrak{X}$ of varieties
(respectively, what classes of fields $k$) is it true that for 
$X \in \mathfrak{X}$, if $X$ admits a $0$-cycle of 
degree one, then $X$ has a rational point.

If $X$ is a curve of genus zero 
or one over any field $k$, then $X$ has a rational  point once it admits 
a zero 
cycles of degree one.  However, the question is more reasonable to ask for 
classes of rational varieties or homogeneous varieties.  In the
setting of rational varieties, there are
examples, due to Colliot-Th\'el\`ene and Coray \cite{colliot_coray}
of conic bundles over the
projective line over a $p$-adic field with a zero cycle of degree one, 
which have no rational points. We shall list from literature some
questions in this  direction for homogeneous varieties.  
{\bf $Q(H P_r)$ (Veisfeiler)\cite{veisf}}
Let $X$ be a projective homogeneous variety under a connected linear algebraic
group defined over a field $k$.  If $X$ has a zero cycle of degree one, does 
$X$ have a rational point?\\
{\bf $Q(PHS)$\cite{serre}}
Let $G$ be a connected linear algebraic group defined over a field $k$.  Let 
$X$ be a principal homogeneous space for $G$  over $k$.  If $X$ has a zero 
cycle of degree one, does  $X$ have a rational point?\\
The following questions combine the above two in a more general setting.\\
{\bf $Q(H)$ (Colliot-Th\'el\`ene)\cite{totaro}}
Let $X$ be a quasi-projective homogeneous variety under a connected 
linear algebraic group defined over $k$.  If $X$ has a zero cycle of degree 
one,  does $X$ have a rational point?\\
{\bf $Q(Hd)$ (Totaro)\cite{totaro}}
Let $X$ be a quasi-projective variety under a connected linear algebraic group
defined over $k$.  If $X$ has a zero cycle of degree $d > 0$, does $X$ have 
a closed point of degree dividing $d$?

Totaro mentions that the most reasonable cases of his question are 
where $X$ is a principal homogeneous space or if $X$ is projective. 

The first example where $Q(H)$ has a negative answer is due to \cite{florence};
the stabilizer of a rational point over the algebraic closure for these
homogeneous spaces is a finite group.

In this paper, we give examples to show that $Q(HPr)$ has a
negative answer in general.\\
\noindent{\bf \S 1 {Connectedness}}\\
\noindent We remark that connectedness is essential with respect to these
questions. 
There exist automorphisms (Coleman automorphisms \cite{coleman}) of finite
groups with the properties:
\begin{enumerate}
\item $f$ is not inner.
\item for every sylow subgroup $H$ of $G$, $f|_H: H \to G$ is given by 
$f(x) = y_H x y^{-1}_H$ for some $y_H \in G$.
\end{enumerate}

The class of $[f]$ in $H^1(G, G)$ for the trivial action of $G$ on $G$ is 
non-trivial, but restricted to each $p$-sylow subgroup, it is trivial. We 
thank P. Gille for bringing to our attention these examples.\\ 
\noindent{\bf \S 2 {Principal homogeneous spaces}}\\
\noindent The case of $Q(PHS)$ is wide open, and in special cases $Q(PHS)$
is proved to have an affirmative answer. The cases of $PGL_n$ and $O_n$
are classical; 
for $O_n$ the result goes back to a theorem of Springer \cite{springer}.
The case of 
unitary groups is settled in the affirmative by Eva-Bayer and Lenstra
\cite{bayer_lenstra}. 
A positive answer to $Q(PHS)$ when $k$ is a number field 
is due to Sansuc \cite{sansuc}. A main ingredient in
the proof is the Hasse principle for principal
homogeneous spaces under semisimple simply
connected linear algebraic groups defined over
number fields. There are conjectures concerning
Hasse principle for fields of virtual cohomological
dimension 2, due to Colliot-Th\'{e}l\`{e}ne 
which have been proved for classical groups \cite{bayer_parimala}.
One can show that $Q(PHS)$ has an affirmative
answer for principal homogeneous spaces under connected
linear algebraic groups defined over a field of virtual
cohomological dimension 2, provided the corresponding
simply connected group satisfies Hasse Principle conjecture
\cite{p1}\\
\noindent{\bf \S 3 {Projective homogeneous varieties}}\\
The question $Q(HPr)$
also admits a positive answer for number fields 
and this can be deduced from the theorem of Harder
on Hasse principle for projective homogeneous
varieties defined over number fields \cite{harder}. 
If one follows the proof of Borovoi of Harder's theorem,
using the non-abelian $H^2$ of Springer to study
homogeneous varieties, one can show, using the results
of \cite{cogipa} that if $k$ is a 2-dimensional
strict henselian field, $Q(HPr)$ has a positive
answer. It is good to study $Q(HPr)$ in the
case of 2-dimensional fields. A positive answer can be
derived for classical groups over $C_2$ fields \cite{p2}.\\
\noindent{\bf \S 4 Example }\\
In this section, we construct an example to show that $Q(HPr)$ has a
negative answer in
general. This example is a refinement of an example given
in \cite{pss}.

Let $k$ be a $p$-adic field containing a primitive $p$-th root of unity
$\xi$ with $p \geq 5$.  Let $K=k ((t))$.  Let $\ell |k$ be a degree two extension which is
totally ramified over $k$. Then $|\ell^*/\ell^{*p}| > |k^*/k^{*p}|$.
Let $L= \ell ((t))$. Let $\mu \in \ell^*$ be
such that $[\mu] \in {\rm ker}(N_{\ell/k}:\ell^* / \ell^{*p} \to
k^*/k^{*p})$ and $[\mu] \neq 1 $ in $\ell^*/\ell^{*p}$.  Let $D$
be the cyclic algebra of degree $p$ over $L$ defined by:
$$X^p = \mu, Y^p = t, XY = \xi YX.$$ 
It is represented by $(\mu) \cup (t) \in H^2 (L, \mu_p)$.  We have, by choice, 
cores$_{L/K} (D)=1$ so that $D$ supports an involution of second kind. Let
$\tau$ be an $L|K$ involution on $D$.  Let $\lambda \in k^*$ be such that 
$\lambda \notin N_{\ell/k} (\ell^*)$.  Let $h$ be the rank 3 hermitian
form $\langle 1, - \lambda, t \rangle$ over $(D, \tau)$.  Then we have 
the following:\\
{\bf Lemma}
The hermitian form $h$ is anisotropic over $(D, \tau)$.\\
{\bf Proof}
Let $\Delta  =\big{\{}a \in D: {\rm Nrd}_{D|L}(a) \in l[[t]]\big{\}}$
be the unique maximal $\ell[[t]]$-order in $D$.
Every element $a$ of $\Delta$ can be written as $a = \pi^n b$,
where $b$ is a unit in $\Delta$ and $\pi$ a generator of the
unique maximal right ideal in $\Delta$. \\
Suppose there exist $v_1,v_2,v_3 \in D$, not all zero,  such that:
\begin{align}
v_1 \tau(v_1) - \lambda v_2 \tau(v_2) + t v_3 \tau(v_3) = 0 \label{iso1}
\end{align}
Without loss of generality, we may assume that each $v_i \in \Delta$.
We write $v_i = \pi^{n_i}u_i$, where $u_i$ is a unit in $\Delta$ for $i = 1, 2, 3$. 
Thus \eqref{iso1} becomes:
\begin{align}
\pi^{n_1}u_1 \tau(\pi^{n_1}u_1) - \lambda \pi^{n_2}u_2 \tau(\pi^{n_2}u_2) + 
t \pi^{n_3}u_3 \tau(\pi^{n_3}u_3) = 0 \label{iso2}
\end{align}
We first consider the case when $n_1$ is smallest of all $n_i$.
In this case setting $m_2 = n_2 - n_1$ and $m_3 = n_3 - n_1$,
\eqref{iso2} can be rewritten as:
\begin{align}
u_1 \tau(u_1) = \lambda \pi^{m_2}u_2 \tau(\pi^{m_2}u_2) 
-t \pi^{m_3}u_3 \tau(\pi^{m_3}u_3) \label{iso3}
\end{align}
Since $t = \pi^p u_0$, $u_0$ is a unit in $\Delta$ and $p$ odd,
the valuation of $\lambda \pi^{m_2}u_2 \tau(\pi^{m_2}u_2)
- t \pi^{m_3}u_3 \tau(\pi^{m_3}u_3)$ is the minimum of
$\{2m_2, p + 2m_3\}$, which is zero; this implies that
$m_2 = 0$. Thus

\begin{align}
\lambda u_2 \tau(u_2) = u_1 \tau(u_1) \left(1 + 
t \tau(u_1)^{-1}u_1^{-1}\pi^{m_3} u_3 \tau(\pi^{m_3}u_3)\right) \label{iso4}
\end{align}
Set $w = \tau(u_1)^{-1}u_1^{-1}\pi^{m_3} u_3 \tau(\pi^{m_3}u_3)$.
Then taking reduced norm on the both sides of \eqref{iso4} we get:
$${\rm Nrd}_{D}\left(\lambda u_2 \tau(u_2)\right)
= {\rm Nrd}_{D}\left(u_1 \tau(u_1)\right){\rm Nrd}_{D}(1+tw)$$
which gives:
$$\lambda^p N_{L|K}\left({\rm Nrd_D}(u_2)\right)
= N_{L|K}\left({\rm Nrd_D}(u_1)\right){\rm Nrd}_{D}(1+tw)$$
Reading the above equality modulo $t$, we conclude that 
$\lambda^p \in N_{\ell|k}(\ell^*)$. Since $\ell$ is a quadratic extension
over $k$ and $p$ is odd, this implies that $\lambda \in  N_{\ell|k}(\ell^*)$.
But this is a contradiction to the choice of $\lambda$ and therefore
the form $h$ is anisotropic in this case. The case when $n_2$ is smallest 
can be treated in a similar manner.

Now we consider the case when $n_3$ is the smallest
among all $n_i$'s. Let $r_1 = n_1 - n_3$ and $r_2 = n_2 - n_3$.
Then \eqref{iso2} becomes:
\begin{align}
\pi^{r_1}u_1 \tau(\pi^{r_1}u_1) - \lambda \pi^{r_2}u_2 \tau(\pi^{r_2}u_2) =
-t u_3 \tau(u_3) \label{iso5}
\end{align}

Suppose $r_1 \neq r_2$. Valuation of $\pi^{r_1}u_1 \tau(\pi^{r_1}u_1)
- \lambda \pi^{r_2}u_2 \tau(\pi^{r_2}u_2)$ is minimum of $\{2r_1,2r_2\}$,
which is
even, while the valuation of $t u_3 \tau(u_3)$ is $p$ which is odd leading
to a contradiction.
Therefore $r_1 = r_2$ and we have
\begin{align}
u_1 \tau(u_1) = \lambda u_2 \tau(u_2)
-t \pi^{-r_1}u_3 \tau(\pi^{-r_1}u_3) \label{iso7}
\end{align}

If $p < 2r_1$, $u_3 \tau(u_3) = t^{-1}\pi^{r_1}\left(-u_1 \tau(u_1)
+ \lambda u_2 \tau(u_2)\right)\tau(\pi^{r_1})$
with the valuation of right hand side positive, leading to a contradiction.
Thus $p > 2r_1$. Taking reduced norm on the both sides of \eqref{iso7} 
and then reading modulo $t$ we conclude, as before
that $\lambda \in N_{\ell|k}(\ell^*)$, which is a contradiction.
Thus the hermitian form $h$ is anisotropic.
\hfill $\square$\\

\noindent{\bf Remark (i)} The algebra $D_{K \left(\left(\sqrt{\lambda}\right)\right)}$
is division and 
$h_{K\left(\sqrt{\lambda}\right)} \simeq \langle 1,-1, t \rangle$ is 
isotropic over
$\left(D_{K\left(\sqrt{\lambda}\right)}, 
\tau_{K\left(\sqrt{\lambda}\right)}\right)$ with a rank one isotropic
subspace (over $\left(D_{K\left(\sqrt{\lambda}\right)}\right)$).\\
{\bf (ii)} Let $M=K (t^{1/p})$.  Then $[M:K] = p, D_M$ is split and $h_M$
is Morita equivalent to a 
$3p$-dimensional hermitian form $\tilde{h}_M$ over $LM|M$.  Since $M$ is a Laurent
series field over a $p$-adic field, every $9$-dimensional quadratic form
over $M$ is isotropic and every $5$-dimensional hermitian form over 
$LM|M$ is isotropic.  Since $p \geq 5, \tilde{h}_M$ has a totally isotropic
subspace of rank at least $p$ over $LM$; hence $h_M$ has a rank one isotropic
subspace over $D_M$.\\

Let $X$ be the variety of rank one (rank over $D$) 
zero subspaces of $h$ over $(D,\tau)$. Then $X$ is a projective variety
homogeneous under the action of $SU(h)$. Further $X(K) = \phi, 
X\left(K\left(\sqrt{\lambda}\right)\right) \neq \phi$ and $X(M) \neq \phi$.
Thus $X$ admits a zero-cycle of degree one and $X$ has no $K$-rational
point.\\ 
{\bf Remark} One may replace the Laurent series fields
over p-adic fields in the above examples by the rational
function field in one variable over p-adic fields.
One needs to use results of \cite{HV} and \cite{PS} stating
that quadratic forms over such fields in sufficiently
many variables have a nontrivial zero.

We shall now describe the parabolic subgroup defining the stabilizer
of a rational point of $X(\bar{K})$.

Let $\triangle$ be the set of simple roots with respect to a pair $(T,B)$
for $G({\overline{k}}) = SU(h)({\overline{k}})$ 
for a choice of a maximal torus $T$
and a Borel subgroup $B$ containing $T$.  
Let $S=\triangle \backslash \{p,2p\}$
where the ordering on vertices of $\triangle$ 
are as in \cite{tits}.
Let $X$ be the variety of 
parabolic subgroups of $G({\overline{k}})$ 
defined by the conjugacy class associated to $S$. 
Since $S$ is invariant under Galois 
action, $X$ is defined over $k$.
We have the following criterion due to Tits \cite{tits} for the existence of
a rational point for $X$:
$X (E) \neq \phi$ for any extension $E|K$ if and only if $S$
contains all the non-distinguished vertices of the Tits index of
$SU(h)(E)$; this is equivalent to $h$ having a rank $1$ isotropic subspace
over $(D_E,\tau_E)$. The variety $X$ is the variety of rank $1$ isotropic
subspaces in $h$.
We have the following Tits indices for $SU (h)_{K\left(\sqrt{\lambda}\right)}$
and $SU (h)_M$ (Witt index of $h$  over M is $p+r$).
\newpage
\begin{figure}[h]

\begin{center}
\resizebox{3.75in}{!}{\includegraphics{148.files/tits1.eps}}
\end{center}
\end{figure}
\begin{center}
{{\bf Tits index for $SU (h)_{K\left(\sqrt{\lambda}\right)}$}}
\end{center}

\begin{figure}[h]

\begin{center}
\resizebox{3.75in}{!}{\includegraphics{148.files/tits2.eps}}
\end{center}

\end{figure}


\begin{center}
{{\bf Tits index for $SU(h)_{M}$}}
\end{center}

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Homi Bhabha Road, Mumbai 400 005, India.


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