% The comments in the beamer preamble not marked with my initials come
% from the beamer template that comes with the beamer installation. -- EWH

\documentclass[handout]{beamer}
% The "[handout]" tells LaTeX to just give the final slide of 
% each overlay.  Remove it to get the version of the slides that
% I used in the actual talk.  -- EWH

\mode<presentation>
{
\usetheme{default}
  % or ...

% I think that Warsaw is a decent theme, but this time I tried
% specifying the inner- and outer-themes myself.  -- EWH

\setbeamercovered{transparent=40}
  % or whatever (possibly just delete it)

% This makes "hidden" items visible, but shaded.  -- EWH

}


\usecolortheme{rose}
\usecolortheme{whale}
% Color schemes that aren't too distracting.  -- EWH

\useoutertheme{split}
\useinnertheme[shadow=true]{rounded}
% Minimal inner and outer themes.  -- EWH

\setbeamertemplate{navigation symbols}{}
% This supresses the stupid navigation symbols in the lower
% right of the slide. -- EWH


\usepackage{amsmath}
%\usepackage[all]{xy}
\usepackage[english]{babel}
% or whatever
\usepackage[latin1]{inputenc}
% or whatever

\usepackage{times}
\usepackage[T1]{fontenc}
% Or whatever. Note that the encoding and the font should match. If T1
% does not look nice, try deleting the line with the fontenc.


\title[Jacobians in isogeny classes of supersingular surfaces]
{Jacobians in isogeny classes of supersingular abelian surfaces over finite fields}
% The title in brackets is the short title, used (e.g.) at the bottom of slides. -- EWH

\author%[Howe, Nart, Ritzenthaler] % (optional, use only with lots of authors)
{Everett W. Howe\inst{1} \and Enric Nart\inst{2} \and Christophe Ritzenthaler\inst{3}}
% - Give the names in the same order as the appear in the paper.
% - Use the \inst{?} command only if the authors have different
%   affiliation.

\institute[CCR, UAB, IML] % (optional, but mostly needed)
{
  \inst{1}%
  Center for Communications Research, La Jolla
  \and
  \inst{2}%
  Universitat Aut\`onoma de Barcelona
  \and
  \inst{3}
  Institut de Math\'ematiques de Luminy
  }
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.

\date[AMS/SF 2006] % (optional, should be abbreviation of conference name)
{AMS sectional meeting, San Francisco\\ 30 April 2006\\ Corrected slides}
% - Either use conference name or its abbreviation.
% - Not really informative to the audience, more for people (including
%   yourself) who are reading the slides online

%\subject{Theoretical Computer Science}
% This is only inserted into the PDF information catalog. Can be left
% out. 



% If you have a file called "university-logo-filename.xxx", where xxx
% is a graphic format that can be processed by latex or pdflatex,
% resp., then you can add a logo as follows:

%\pgfdeclareimage[height=0.5cm]{university-logo}{idalogoblack}
%\logo{\pgfuseimage{university-logo}}



% Delete this, if you do not want the table of contents to pop up at
% the beginning of each subsection:
\AtBeginSubsection[]
{
  \begin{frame}<beamer>
    \frametitle{Outline}
    \tableofcontents[currentsection,currentsubsection]
  \end{frame}
}


% If you wish to uncover everything in a step-wise fashion, uncomment
% the following command: 

%\beamerdefaultoverlayspecification{<+->}


% That's the end of the beamer preamble.  Now it's just regular LaTeX stuff...  --- EWH

%% Theorem environments.

\newtheorem{proposition}{Proposition}
\theoremstyle{remark}
\newtheorem*{question}{Question}
\newtheorem*{answer}{Answer}
\newtheorem*{heuristic}{Heuristic}
\newtheorem*{exercise}{Exercise}

%% Special characters -- bold face.
\newcommand{\BF}{\mathbb{F}}
\newcommand{\BFbar}{\overline{\mathbb{F}}}
\newcommand{\BH}{\mathbb{H}}
\newcommand{\BP}{\mathbb{P}}
\newcommand{\BQ}{\mathbb{Q}}
\newcommand{\BZ}{\mathbb{Z}}

%% Special characters -- calligraphic.
\def\calO{\mathcal{O}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}

%% Special characters -- fraktur.
\def\frakp{\mathfrak{p}}
\def\frakq{\mathfrak{q}}

%% Math operators.
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\Gal}{Gal}
\DeclareMathOperator{\Jac}{Jac}
\DeclareMathOperator{\trace}{trace}



\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Title page
\begin{frame}
  \titlepage
\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Page 1
\begin{frame}
\frametitle{Weil polynomials of elliptic curves}

\begin{block}{Elliptic curves over finite fields.}
$$\left\{\vcenter{\hsize=1.5in\noindent
  \hfill Isogeny classes of\hfill\break
  \phantom{.}elliptic curves over $\BF_q$}\right\}
\hookrightarrow\big\{x^2 - tx + q\big\}$$
\end{block}
\pause\medskip
The image is known (Deuring, Honda-Tate, Waterhouse).

\pause\medskip
Suppose $q$ is a power of a prime $p$.

\begin{block}{The possible values of $t$:}
\begin{itemize}
\item Every $t$ with $(t,q)=1$ and $t^2 < 4q$.
\item If $q$ is not a square: $\BZ\cap\{0,\pm\sqrt{2q},\pm\sqrt{3q}\}$.
\item If $q$ is a square: $\pm2\sqrt{q}$,
\item[] \hskip 1.18in $\pm\sqrt{q}$ \ (if $p\not\equiv 1\bmod 3$),
\item[] \hskip 1.44in $0$ \ (if $p\not\equiv 1\bmod 4$).
\end{itemize}
\end{block}


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Page 2
\begin{frame}
\frametitle{Weil polynomials of genus-$2$ curves}

% Yes, the following block is a kludge.
\begin{block}{Genus-$2$ curves over finite fields.}
\begin{align*}
\left\{\vcenter{\hsize=1.7in\noindent
\hfill Isogeny classes of\hfill\break
\phantom{.}abelian surfaces over $\BF_q$}\right\}
&\leftrightarrow
&
\vcenter{\hsize=1.75in\noindent
\hfill Known subset of\hfill\break
$\big\{x^4 + ax^3 + bx^2 + aqx + q^2\big\}$}\\
\pause
\vcenter{\hsize=1.7in\noindent\qquad\quad\quad$\bigcup$\hfill}& &
\vcenter{\hsize=1.7in\noindent\qquad\quad\quad\,\ $\bigcup$}\\
\left\{\vcenter{\hsize=1.7in\noindent
\hfill Isogeny classes that\hfill\break
\phantom{.}\hfill contain Jacobians\hfill\phantom{.}}\right\}
&\leftrightarrow
&\vcenter{\hsize=1.7in\noindent\quad\quad\quad\,\ \ \  ???}\\
\end{align*}
\end{block}

\pause
Honda-Tate tells us what the upper right set is.

\pause\bigskip
R\"uck (1990) asked for the image of the lower left in the 
upper right.  Sixteen years later, the answer is known. 

\pause\bigskip
Contributions by
Adleman, EWH, Huang, Lauter, Maisner, McGuire, Nart,
Ritzenthaler, R\"uck, Serre, Voloch, $\ldots$


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Page 3
\begin{frame}
\frametitle{The contents of this talk}
As elliptic curve examples suggest, special cases arise
in analysis of supersingular isogeny classes.
We'll look at one.
\pause\bigskip

I'll sketch a proof of the `only if' part of the following theorem.

\pause
\begin{theorem}
Suppose $E_1$ and $E_2$ are supersingular elliptic curves over 
a finite field of characteristic greater than $3$.  Then there
is a Jacobian isogenous to $E_1\times E_2$ if and only if
$$\trace E_1 = \pm\trace E_2.$$
\end{theorem}
\pause

The assumption on the characteristic is necessary.\hfill\break
\pause
In characteristic $3$, \emph{neither} implication is true.

\pause\bigskip
The proof relies on the structure of the $p$-torsion subgroup-schemes
of supersingular elliptic curves.  


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Page 4
\begin{frame}
\frametitle{Different isogeny classes have different $p$-torsion}

\begin{proposition}
If $E_1$ and $E_2$ are non-isogenous supersingular elliptic curves 
over $\BF_q$ {\rm(}in characteristic $>3${\rm)},
then $E_1[p] \not\cong E_2[p].$
\end{proposition}

\pause\bigskip

In fact, we will see that isogenous curves may have 
non-isomorphic $p$-torsion groups.

\pause\bigskip
We'll see later how this proposition will help us prove the theorem
about Jacobians isogenous to $E_1 \times E_2$.


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Page 5
\begin{frame}
\frametitle{The isogeny classes we must consider}
Suppose $E_1$ and $E_2$ are supersingular 
elliptic curves over $\BF_q$ with $\trace E_1\neq\pm\trace E_2$.
\pause\bigskip

In characteristic $p>3$, this implies $q = \text{(square)}$.
\pause\bigskip

\begin{block}{Possible supersingular traces over $\BF_q$ in this case.}
\begin{tabular}{r|l|l}
trace        & condition on $p$       & size of isogeny class\\ \hline
$-2\sqrt{q}$ &                        & $\lfloor (p+4)/6\rfloor - \lfloor p/12\rfloor$\\
$ -\sqrt{q}$ & $p\not\equiv 1\bmod 3$ & $2$ \\
0            & $p\not\equiv 1\bmod 4$ & $2$ \\
$  \sqrt{q}$ & $p\not\equiv 1\bmod 3$ & $2$ \\
$ 2\sqrt{q}$ &                        & $\lfloor (p+4)/6\rfloor - \lfloor p/12\rfloor$\\
\end{tabular}
\end{block}



\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Page 6
\begin{frame}
\frametitle{Twists of $p$-divisible groups}
Let $T_p E$ denote the $p$-divisible group of $E$.

\pause\bigskip
Waterhouse: All $E$ with trace $2\sqrt{q}$ have same $T_p E$.
Let $M$ be this $p$-divisible group, and $M_0$
its $p$-torsion.

\pause\bigskip
Every supersingular EC over $\BF_q$ has a twist with trace $2\sqrt{q}$.

\pause\bigskip
We will show that if $\trace E = 2\sqrt{q}$ then
$$\begin{matrix}
H^1(G_{\BF_q}, \Aut E)&  \hookrightarrow&  H^1(G_{\BF_q}, \Aut M)&
  \hookrightarrow&  H^1(G_{\BF_q}, \Aut M_0)\\
\parallel&&\parallel&&\\
(\mu_2,\mu_4,\text{ or } \mu_6) & \hookrightarrow & \BF_{p^2}^* & &
\end{matrix}$$
where $G_{\BF_q} = \Gal(\BFbar_q/\BF_q)$.
  

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Page 7
\begin{frame}
\frametitle{Enumerating the twists}
Suppose $E$ has trace $2\sqrt{q}$.  Tate showed that
$$\End T_p E = (\End E)\otimes\BZ_p = 
\text{maximal order $\calO$ in $\BH_p$}.$$
\pause
Then $\calO$ contains the ring $R$ of Witt vectors over $\BF_{p^2}$, and
\begin{align*}
\text{twists of $T_p E$} 
  &\leftrightarrow H^1(G_{\BF_q}, \Aut T_p E)\\
\pause  &\leftrightarrow \text{conj.~classes of elts.~of $\calO$ of finite order}\\
\pause  &\leftrightarrow \text{roots of unity in $R\subset \calO$}\\
\pause  &\leftrightarrow \text{elements of $\BF_{p^2}^*$.}
\end{align*}  
\pause
Not hard to  write down the Dieudonn\'e modules of
these twists, and see they are all distinct mod $p$.
So 
$$\only<7>{H^1(G_{\BF_q}, \Aut E) \hookrightarrow}
           H^1(G_{\BF_q}, \Aut M) \hookrightarrow 
           H^1(G_{\BF_q}, \Aut M_0).$$
 


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Page 8
\begin{frame}
\frametitle{Split Jacobians and torsion subgroups}

\begin{lemma}
Let $E_1$ and $E_2$ be elliptic curves over $\BF_q$.

Suppose $C/\BF_q$ satisfies $\Jac C \sim E_1\times E_2$.  

Then there are elliptic curves $F_1\sim E_1$ and $F_2\sim E_2$
and an integer $n>1$
such that $F_1[n]\cong F_2[n]$.\qed
\end{lemma}

\pause
Note that then 
\alert{$$\trace E_1 \equiv \trace E_2\bmod n.$$}

\pause\medskip
(In our paper we use a much stronger version, due to Kani.)

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}
%\frametitle{Sketch of proof}
%Write $$0\to\Delta\to E_1\times E_2\to\Jac C\to 0.$$
%\pause
%Find the largest $\Delta_1\times\Delta_2\subset\Delta$, and
%let $F_i = E_i/\Delta_i$ and $\Delta' = \Delta/(\Delta_1\times\Delta_2)$.
%Then we have
%$$0\to\Delta'\to F_1\times F_2\to\Jac C\to 0$$
%but now $\Delta'\hookrightarrow F_1$
%and $\Delta'\hookrightarrow F_2$.
%
%\pause
%
%Pulling back the principal polarization of $\Jac C$ to
%$F_1\times F_2$ gives a polarization $(m,n)$ on $F_1\times F_2$
%of degree $(\#\Delta')^2$.  Then
%$$F_1[m] \cong \Delta' \cong F_2[n].$$
%
%\pause
%We can't have $m=n=1$, for then $\Jac C$ would be a 
%product of polarized elliptic curves. \qed
%
%
%\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Page 9
\begin{frame}
\frametitle{Proof of `only if' part of main theorem}

Suppose that $\trace E_1 \neq \pm\trace E_2$ and 
that there is a $C$ with $\Jac C\sim E_1\times E_2$.

\pause\bigskip

The lemma says that we have $F_1[n]\cong F_2[n]$ for some divisor $n>1$ 
of $\trace E_1 - \trace E_2$ and some $F_1\sim E_1$, $F_2\sim E_2$.

\pause\bigskip
The proposition shows that $n$ cannot be divisible by $p$.

\pause\bigskip

The gives a contradiction when $|\trace E_1 - \trace E_2| = \sqrt{q}$.
\pause\bigskip

In the remaining cases, one of the $E$'s (say $E_1$) has trace $\pm2\sqrt{q}$ and the
other does not, and the $n$ from the lemma is $2$ or $3$.

\pause\bigskip
But Frobenius acts as an integer on $F_1[2]$ and $F_1[3]$ for every 
$F_1\sim E_1$, while it does not do so for any $F_2\sim E_2$.
\qed

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Page 10
\begin{frame}
\frametitle{Where's the computation?}

Nart, Ritzenthaler, and I used computer calculations of 
Weil polynomials of supersingular curves to help determine
what we should be proving.

\pause\bigskip

Most surprising result, that took the most work to prove:

\pause\bigskip
\begin{theorem}
Let $q$ be even power of a prime $p\not\equiv 1 \bmod 12$,
so that $x^4 - qx^2 + q^2$ is the Weil polynomial of an
abelian surface.  Then there is a curve with this Weil 
polynomial if and only if $p\not\equiv 11\bmod 12$
and $p\neq3$.
\end{theorem}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Page 11
\begin{frame}
\frametitle{Notes added after the talk}

Ren\'e Schoof asked why we computed the twists of $M$, when we
only needed the twists of $M_0$.  

\medskip

One reason is that the automorphism group of $M$ is `nicer'
than that of $M_0$, so that the calculation of the $H^1$
seems a little cleaner.

\medskip

But it also just seemed like a natural thing to do.

\medskip

I didn't mention it in the talk, but in fact
$H^1(G_{\BF_q},\Aut M_0)$ is also isomorphic to $\BF_{p^2}^*$.

\end{frame}






\end{document}







