\documentclass[12pt]{amsart} \usepackage{amsmath,amsthm,amsfonts}
\usepackage[all]{xy}
\CompileMatrices

\newcommand{\G}{{\mathbb G}} \newcommand{\C}{{\mathbb C}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\D}{{\mathbb D}}
 \newcommand{\Qbar}{{\overline{\Q}}}
\newcommand{\kbar}{{\overline{k}}} \newcommand{\Z}{{\mathbb Z}}
\newcommand{\N}{{\mathbb N}} \newcommand{\R}{{\mathbb R}}
\newcommand{\Aff}{{\mathbb A}} \newcommand{\Proj}{{\mathbf P}}
\newcommand{\OO}{{\mathcal O}} \newcommand{\TT}{{\mathcal T}}
\newcommand{\BB}{{\mathcal B}} \newcommand{\DD}{{\mathcal D}}
\newcommand{\EE}{{\mathcal E}} \newcommand{\CC}{{\mathcal C}}
\newcommand{\AAA}{D_k} \newcommand{\pp}{{\mathfrak p}}
\newcommand{\PP}{{\mathfrak P}} \newcommand{\qq}{{\mathfrak q}}
\newcommand{\QQ}{{\mathfrak Q}} \newcommand{\mm}{{\mathfrak m}}
\newcommand{\cc}{{\mathfrak c}} \newcommand{\ttt}{{\mathfrak t}}
\newcommand{\aaa}{{\mathfrak a}} \newcommand{\F}{{\mathbb F}}
 \newcommand{\FF}{{\mathcal F}}
 \newcommand{\VV}{{\mathcal V}}
\newcommand{\ZZ}{{\mathcal Z}}
 \newcommand{\pibar}{{\overline{\pi}}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\Card}{\operatorname{Card}}
\newcommand{\Gal}{\operatorname{Gal}}
\newcommand{\inv}{\operatorname{inv}}
\newcommand{\End}{\operatorname{End}}
\newcommand{\Endo}{\operatorname{End}_k^0}
\newcommand{\GalQ}{{\Gal}(\Qbar/\Q)}
\newcommand{\defequal}{\stackrel{\text{def}}{=}}
\newcommand{\isom}{\cong} \newcommand{\tensor}{\otimes}
\newcommand{\tors}{{\operatorname{tors}}} \newcommand{\directsum}{\oplus}
\newcommand{\ord}{{\operatorname{ord}}}
\newcommand{\Br}{{\operatorname{Br}}}
\newcommand{\Res}{{\operatorname{Res}}}
\newcommand{\Norm}{{\bf{{\operatorname{N}}}}}
\newcommand{\Ind}{{\operatorname{Ind}}}
\newcommand{\INT}{\operatorname{INT}} \title[The theorem
of Honda and Tate]{The theorem of Honda and Tate}
\author{Kirsten Eisentr\"ager}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{cor}[theorem]{Corollary}
\newtheorem{prop}[theorem]{Proposition}
\newtheorem{notation}[theorem]{Notation} \theoremstyle{definition}
\newtheorem{question}{Question$\!\!$} \renewcommand{\thequestion}{}
\newtheorem{definition}[theorem]{Definition} \theoremstyle{remark}
\newtheorem{rem}{Remark$\!\!$} \renewcommand{\therem}{} 

\newcommand{\marginalfootnote}[1]{%
        \footnote{#1}
        \marginpar[\hfill{\sf\thefootnote}]{{\sf\thefootnote}}}
\newcommand{\edit}[1]{\marginalfootnote{#1}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\numberwithin{equation}{section}
\begin{document}
%\begin{abstract}
%  \end{abstract}
\maketitle
\section{Motivation}
Consider the following theorem of Tate
(proved in \cite[Theorems~2--3, Appendix I]{Mumford}):
\begin{theorem}\label{motivation}
  Let $A$ and $B$ be abelian varieties over a finite field $k$ of size
  $q$, and let $f_A$, $f_B \in \Z[T]$ be the characteristic polynomials
  of their $q$-Frobenius endomorphisms, so $f_A$ and $f_B$ have
  degrees $2 \cdot {\rm{dim}} A$ and $2\cdot {\rm{dim}} B$ with constant
  terms $q^{{\rm{dim}} A}$ and $q^{{\rm{dim}} B}$ respectively. The
  following statements are equivalent:

\begin{enumerate}
\item $B$ is $k$-isogenous to an abelian subvariety of $A$.
\item $f_B|f_A$ in $\Q[T]$.
\end{enumerate}
In particular, $A$ is $k$-isogenous to $B$ if and only if $f_A = f_B$.
Moreover, $A$ is $k$-simple if and only if $f_A$ is a power
of an irreducible polynomial in $\Q[T]$.

\end{theorem}
Recall that the roots of $f_A$ in $\C$ are ``Weil $q$-integers'':
algebraic integers whose images in $\C$ all have absolute value
$q^{1/2}$.  By Theorem~\ref{motivation} to describe the isogeny
classes of abelian varieties over finite fields we would like to know
for which Weil $q$-integers $\pi$ in $\C$ we can find a $k$-simple
abelian variety $A/k$ such that $\pi$ is a root of $f_A$.  We will show
that for each $\pi$, we can find a $k$-simple abelian variety $A$ and
an embedding $\Q[\pi] \hookrightarrow \End^0_{k}(A)$ such that $\pi$ is
the $q$-Frobenius endomorphism of $A$.  We will follow the discussion
in \cite{IsogenyClasses} and \cite{MiWa}. Brian Conrad heavily revised
Section~\ref{sec8} to simplify the computation of the invariant
$\inv_v(E)$ for $v \mid p$ and he provided the proofs for
Theorem \ref{analogue} and the results in the appendix.

\section{Setup}\label{setup}
Let $k$ be a finite field with $q=p^a$ elements. We will work in the
{\it category $M(k)$
of abelian varieties up to isogeny} over $k$: $M(k)$ is
the category whose objects are abelian varieties over $k$ with the set
of morphisms between two abelian varieties $A$ and $B$ given by
$\Hom_k(A,B) \tensor \Q$.  Two abelian varieties are isomorphic in
$M(k)$ if and only if they are isogenous over $k$.  The category
$M(k)$ is a $\Q$-linear category.

Our goal is to describe the category $M(k)$. Since every object in
$M(k)$ is a direct sum of a finite number of simple objects, it
suffices to list the isomorphism classes of simple objects and, for
each class, the endomorphism algebra. The theorems of Honda and Tate
allow us to do this, as we shall see.

{\bf Notation:} For an abelian variety $A$ over $k$, $\pi_A$ will
denote the $q$-Frobenius endomorphism. 
For abelian varieties $A$ and $B$ we use $\Hom^{0}_k(A,B)$ to denote
$\Hom_k(A,B) \tensor \Q$.

\medskip

Let $A$ be an abelian variety over $k$, so $\pi_A$ commutes with all
endomorphism of $A$ and hence lies in the center of $\Endo(A)$.  If $A$
is simple, then $\Endo (A)$ is a division algebra. Therefore, when $A$
is simple the subring $\Q[\pi]$ is a number field. An isogeny $A \to B$
of simple abelian varieties over $k$ defines an isomorphism $\Endo (A)
\to \Endo (B)$ carrying $\pi_A$ into $\pi_B$ and hence mapping $\Q[\pi_A]$
isomorphically onto $\Q[\pi_B]$. We also know that $\pi_A$ is an
algebraic integer whose absolute value is $q^{1/2}$ under every
embedding into $\C$. This motivates the following definition.
\begin{definition}
A {\em Weil $q$-integer} in a field
of characteristic $0$
is an algebraic integer $\pi$ such that for
every embedding $\sigma: \Q[\pi]\hookrightarrow \C$, $|\sigma \pi|=q^{1/2}$.
\end{definition}

Let $W(q)$ be the set of Weil $q$-integers in $\C$.
\begin{definition}
Two Weil $q$-integers $\pi$ and $\pi'$ are {\em conjugate}, $\pi \sim
\pi'$, if there exists an isomorphism of $\Q[\pi] \to
\Q[\pi']$ carrying $\pi$ into $\pi'$.
\end{definition}
Equivalently, $\pi$ and $\pi'$ have the same minimal polynomial over
$\Q$.

Let $A$ be a simple abelian variety defined over $k$.
We will describe the division algebra $\Endo (A)$ in terms of
$\pi_A$. To do this, we first need to review the classification of
division algebras over number fields.

\section{Review: division algebras over global fields}\label{revcsa}

\begin{definition}
A {\em central simple algebra} over a field $F$ is an
$F$-algebra $R$ such that
\begin{enumerate}
\item $R$ is finite dimensional over $F$.
\item $F$ is the center of $R$.
\item $R$ is a simple ring.
\end{enumerate} 
\end{definition}
If $R$ is also a division algebra, we call it a {\em central division
  algebra} over $F$.\\
{\bf Notation:} We will denote the $r \times r$ matrix algebra over a
field $F$ by $M_r(F)$.
\begin{lemma}
If $R$ and $S$ are central simple algebras over $F$, then so is
$R\tensor_F S$. For any field $F'/F$, $R\tensor_F F'$ is a central
simple algebra over $F'$.
\end{lemma}
\begin{proof}
  For the first part see \cite[(7.6) Theorem]{Reiner}, for the
  second statement see \cite[(7.8) Corollary]{Reiner}.
\end{proof}
\begin{prop}[Wedderburn's Theorem]\label{Wedderburn}
  Every central simple algebra over $F$ is isomorphic to 
  ${\rm{End}}_E(E^{\oplus r}) = M_r(E)$ for
  some $r \geq 1$ and some central division algebra $E$ over $F$. Moreover,
  $r$ is uniquely determined by $R$, and $E$ is uniquely determined up
  to $F$-isomorphism.
\end{prop}
\begin{proof}
See \cite[p.\ 91]{Reiner}.
\end{proof}
\begin{definition}[Brauer group]
  Let $A$ and $B$ be finite dimensional central simple algebras over a
  field $F$. We say that $A$ and $B$ are {\em similar}, $A \sim B$, if $A
  \tensor_F M_n(F) \isom M_n(A)$ is $F$-isomorphic
to $B\tensor_F M_m(F) \isom M_m(B)$ for some $m, n \geq 1$.
  Define the {\em Brauer group} of $F$, $\Br(F)$, to be the set of
  similarity classes of central simple algebras over $F$, and write
  $[A]$ for the similarity class of $A$. For classes $[A]$ and $[B]$,
  define
\[
[A][B]:= [A\tensor _F B].
\]
This is well defined and makes $\Br(F)$ into an abelian group with
$[A]^{-1}=A^{\operatorname{opp}}$, where $A^{\operatorname{opp}}$ is
the ``opposite algebra''.
\end{definition}
By Proposition~\ref{Wedderburn}, each element in $\Br(F)$ is
represented (up to isomorphism) by a unique central division algebra over
$F$.  

We now need to clear up any possible confusion concerning
sign conventions for local invariants of Brauer classes
for non-archimedean local fields. 
Upon choosing a separable closure
$F_{\rm{s}}$ of an arbitrary field $F$, 
there are two natural procedures to define
a functorial 
isomorphism ${\rm{Br}}(F) \simeq {\rm{H}}^2(F_{\rm{s}}/F,
F_{\rm{s}}^{\times})$:
a conceptual method via non-abelian cohomology
as in \cite[Ch.~X,~\S5]{Serre} and an explicit method via
crossed-product algebras.  By \cite[Ch.~X,~\S5,~Exer.~2]{Serre},
these procedures are negatives of each other.  We choose
to use the conceptual method of non-abelian cohomology.  In case $F$ is
a non-archimedean local field with residue field $\kappa$ and $F^{\rm{un}}$
denotes its maximal unramified subextension within $F_{\rm{s}}$
(with $\overline{\kappa}$ the residue field of $F^{\rm{un}}$)
it is known from local
class field theory that the natural map
${\rm{H}}^2(F^{\rm{un}}/F,{F^{\rm{un}}}^{\times}) \rightarrow
{\rm{H}}^2(F_{\rm{s}}/F,F_{\rm{s}}^{\times})$ is an isomorphism,
and the normalized valuation mapping
${F^{\rm{un}}}^{\times} \rightarrow \Z$ induces an isomorphism
$${\rm{H}}^2(F^{\rm{un}}/F,{F^{\rm{un}}}^{\times}) \simeq
{\rm{H}}^2(F^{\rm{un}}/F,\Z) 
\stackrel{\delta}{\simeq} {\rm{H}}^1({\rm{Gal}}(F^{\rm{un}}/F),\Q/\Z).$$
There now arises the question of choice of topological
generator for ${\rm{Gal}}(\overline{\kappa}/\kappa)$: arithmetic
or geometric Frobenius?  We choose to follow
Deligne's convention and work with geometric Frobenius;
in \cite[p.~130]{Cassels} and \cite[p.~193]{Serre} the arithmetic Frobenius
generator is used instead.
Via evaluation on
the chosen topological generator, our conventions lead
to a composite isomorphism
$${\rm{inv}}_F:{\rm{Br}}(F) \simeq \Q/\Z$$
for non-archimedean local fields $F$.  If one
uses the arithmetic Frobenius convention, then by adopting
the crossed-product algebra method to define the isomorphism
${\rm{Br}}(F) \simeq {\rm{H}}^2(F_{\rm{s}}/F,F_{\rm{s}}^{\times})$
one gets the {\em same} composite isomorphism
${\rm{inv}}_F$ since the two sign differences
cancel out in the composite.  ({\bf Warning}: In
\cite{Cassels} and \cite{Serre} the Brauer group of $F$ is
{\em defined} to be ${\rm{H}}^2(F_{\rm{s}}/F,F_{\rm{s}}^{\times})$,
and so the issue of choosing between non-abelian cohomology or
crossed-product algebras does not arise in the foundational
aspects of the theory; however, it implicitly arises
in comparison with
central simple algebras,
such as in \cite[pp.~137-8]{Cassels} where the details are omitted.)

Since ${\rm{Br}}(\R)$ is cyclic of order 2 and 
${\rm{Br}}(\C)$ is trivial, for archimedean local fields $F$ there
is a unique injective homomorphism 
${\rm{inv}}_F:{\rm{Br}}(F) \hookrightarrow \Q/\Z$. 

By \cite[Thm.~3,~p.~131]{Cassels}, for a finite extension $F'/F$ of
non-archimedean local fields, composition
with the natural map ${\rm{Br}}(F) \to
{\rm{Br}}(F')$ carries ${\rm{inv}}_{F'}$ to $[F':F]\cdot
{\rm{inv}}_F$.  By \cite[p.~194,~Cor.~3]{Serre}, ${\rm{inv}}_F(E)$ has
order $\sqrt{[E:F]}$ for any central division algebra $E$ over $F$.
These assertions are trivially verified to also hold for archimedean
local fields $F$.

\begin{theorem}%\cite[Remark (ii), p.~277]{Reiner}
\label{exact}
%\cite[Chapter VII, p.~185, p.~195]{Cassels}\label{exact}
  Let $F$ be a global field.  There is an exact sequence
\[
0 \to \Br(F) \to \bigoplus_{ v \in M_{F}} \Br(F_v)
\stackrel{\sum}{\to} \Q /\Z \to 0,
\]
where $M_{F}$ denotes the set of nonequivalent nontrivial absolute
values of $F$ and $F_v$ denotes the completion of $F$ at $v$. The
first map is defined via the maps $\Br(F)\to \Br(F_v)$ given by
extension of scalars, and the second map is given by summing the local
invariants.
\end{theorem}
\begin{proof}
This is \cite[\S9.7,~\S11.2]{Cassels}.
\end{proof}
For a global field $F$ and a central division algebra $E$ over $F$,
write $\inv_v(E)$ for $\inv_{F_v}(E\tensor F_v)$.  The theorem says
that a central division algebra $E$ over a global field $F$ is
uniquely determined up to isomorphism by its invariants $\inv_v(E)$,
and that these may be arbitrarily assigned subject to the condition
$\inv_v(E)=0$ for all but finitely many $v$ and $\sum \inv_v(E)=0$.
Moreover, the order of $[E]$ in $\Br(F)$ is the least common
``denominator'' of the $\inv_v(E) \in \Q/\Z$.

If $K$ is any field then for a class $c \in \Br(K)$ its {\em period} is
its order and its {\em index} is $\sqrt{[\Delta:K]}$ with $\Delta$ the unique
central division algebra over $K$ representing the class $c$.  It is a
classical fact that the period divides that index and that these
integers have the same prime factors (see \cite[X.5]{Serre},
especially Lemma 1 and Exercise 3), but in general equality does not
hold.  (There are function fields of complex 3-folds for which some
order-2 elements in the Brauer group cannot be represented by a
quaternion algebra, and counterexamples with less interesting fields were first
discovered by Brauer.)  We have noted above that over local fields
there is equality of period and index (the archimedean case being
trivial).  We need one more theorem that gives such a result over
global fields:

\begin{theorem}\label{thm3.7}
  For a central division algebra $E$ over a global field $F$, the
  order of $[E]$ in $\Br(F)$ is $\sqrt{[E:F]}$.
\end{theorem}

As a special (and very important) case, elements of order 2 in
${\rm{Br}}(F)$ are precisely the Brauer classes of quaternion division
algebras for a global field $F$; as noted above, this fails for more
general fields.  We shall require this special case for $\Q$ and real
quadratic fields in our study of $k$-simple $A$ for which $f_A$ has a
real root.  Since Theorem \ref{thm3.7} does not seem to be explicitly
stated in any of the standard modern references on class field theory
(though it is cryptically alluded to in \cite[p.~105]{artintate}), we
give a proof below.

\begin{proof}
  Let $E$ have degree $n^2$ over $F$ and let $d$ be the order of $[E]$
  in $\Br(F)$, so $d|n$.  Note that $d$ is the least common multiple
  of the local orders $d_v$ of $[E \otimes_F F_v] \in \Br(F_v)$ for each
  place $v$ of $F$, with $d_v = 1$ for complex $v$, $d_v|2$ for real
  $v$, and $d_v = 1$ for all but finitely many $v$.  Using these
  formal properties of the $d_v$'s, we may call upon the full power of
  global class field theory via Theorem 5 in \cite[Ch.~X]{artintate}
  to infer the existence of a cyclic extension $L/F$ of degree $d$
  such that $[L_{v'}:F_v]$ is a multiple of $d_v$ for every $v$ of $F$
  (here, $v'$ is any place on $L$ over $v$, and the constraint on the
  local degree is only non-trivial for $d_v > 1$).  In the special
  case $d = 2$ (the only case we will require) one only needs weak
  approximation and Krasner's Lemma rather than class field theory:
  take $L$ to split a separable quadratic polynomial over $F$ that
  closely approximates ones that define quadratic separable extensions
  of $F_v$ for each $v$ such that $d_v = 2$.
  
  Since restriction maps on local Brauer groups induce multiplication
  by the local degree on the local invariants, it follows that $E \otimes_F
  L$ is locally split at all places of $L$, whence by the injectivity
  of the map from the global Brauer group into the direct sum of the
  local ones (for $L$) we conclude that the Galois extension $L/F$
  {\em of degree $d$} splits $E$.  (The existence of cyclic splitting
  fields for all Brauer classes is proved for number fields in Tate's
  article in \cite{Cassels} and is proved in general in Weil's ``Basic
  Number Theory'', but neither reference seems to exert much control
  on the degree of the global cyclic extension.)  By a general result
  concerning Brauer groups of arbitrary fields
  \cite[Ch.~X,~\S5,~Lemma~1]{Serre}, every Brauer class split by a
  Galois extension of degree $r$ is represented by a central simple
  algebra with degree $r^2$.  Applying this fact from algebra in our
  situation, $[E] = [C]$ for a central simple algebra $C$ of degree
  $d^2$ over $F$.  But each Brauer class is represented by a unique
  central division algebra, and so $C$ must be $F$-isomorphic to a
  matrix algebra over $E$.  Since $[C:F] = d^2$ and $[E:F] = n^2$ with
  $d|n$, this forces $d = n$ as desired.
\end{proof}


\section{Statement of Theorem}
We will keep the notation of Section~\ref{setup}.
\begin{theorem}\label{theorem}$\left. \right.$
\begin{enumerate}
\item Let $k$ be a finite field of size $q=p^a$.  The map $A \mapsto
  \pi_A$ defines a bijection from the set of $k$-isogeny classes of
  simple abelian varieties over $k$ to the set
  $W(q)/(\text{conjugacy})$.

Let $A \in M(k)$ be simple. Let $E:= \End_k^0(A)$ and let $F:= \Q[\pi_A]
\subseteq E$. Then $E$ is a division algebra whose center is $F$.
\item \label{partii}The division algebra $E$ does not split at any
  real place of $F$, and $E$ splits at all finite places of $F$ not
  dividing $p=\operatorname{char}(k)$.  For a place $v$ of $F$
  dividing $p$ we have
\begin{equation}\label{eq:1}
\inv_v(E) = \frac{v(\pi_A)}{v(q)}\cdot [F_v:\Q_p] \mod 1,
\end{equation}
where $F_v$ denotes the completion of $F$ at $v$.
We have
\begin{equation}\label{eq:2}
2 \cdot \dim A = [E:F]^{1/2}\cdot [F:\Q].
\end{equation}
\end{enumerate}
\end{theorem}
The injectivity of the map $A \mapsto \pi_A$ was proved by Tate, and
the surjectivity of this map 
was proved by Honda~\cite{Honda}. Part(\ref{partii})
of the theorem was proved by Tate in \cite{TateInv}.  An interesting
consequence of (\ref{eq:2}) is that if $K \subseteq E$ is a maximal
commutative subfield over $F$ (so $[K:F] = \sqrt{[E:F]}$) then $[K:\Q]
= 2 {\rm{dim}}A$.  Hence, every $k$-simple abelian variety over a
finite field $k$ is of CM type {\em over $k$}.

\section{Examples}
Let $\pi \in W(q)\subseteq \C$ be a Weil $q$-integer in $\C$. Grant
Theorem~\ref{theorem}.
\subsection{Real Case:}\label{realcase}
Assume $F=\Q[\pi]$ has a real embedding, and fix such an embedding
$\Q[\pi] \stackrel{\phi}{\hookrightarrow}\R \subseteq \C$. Then
$\phi(\pi)=\overline{\phi(\pi)}$.  In this case $\phi(\pi^2)=\phi(\pi) \cdot
\overline{\phi(\pi)} = q = p^a$, so $\pi^2 =p^a$. Of course, $\pm \sqrt{q}$
is a Weil $q$-integer for any $q$, so by Theorem \ref{theorem}(1) it
does arise from a $k$-simple $A$ that is unique up to $k$-isogeny.
There are two possible cases to consider.
\begin{enumerate}
\item When $a$ is even, we have $F = \Q$. There is only one real place
  $v$, and for this place we have $\inv_v(E)=1/2$, which implies that
  $E$ must be ramified at $p$ as well with $\inv_p(E) =1/2$. This
  implies that the order of $E$ in $\Br(\Q)$ is 2, and so (Theorem
  \ref{thm3.7}) $E$ is a quaternion division algebra over $\Q$ which
  is ramified at $\infty$ and $p$. The formula $2 \dim A =
  [E:F]^{1/2}[F:\Q]$ implies that $\dim A =1$.  Hence, the
  characteristic polynomial of $f_A \in \Z[T]$ with a factor $T -
  p^{a/2}$ has degree 2 and constant term $q = p^a$, so it must be $(T
  - p^{a/2})^2$.
  
  Since $a_p(A) = 2 p^{a/2} \in p\Z$, $A$ is a supersingular elliptic
  curve.  For any abelian variety $X$ over any field $K$, the
  faithfulness of $V_{\ell}(X)$ as a $\Q_{\ell} \otimes
  {\rm{End}}_K^0(X)$-module for any prime $\ell \neq {\rm{char}}(K)$
  ensures that $\dim_{\Q} {\rm{End}}_K^0(X) \leq (2 {\rm{dim}}X)^2$.
  Thus, for $\Q$-dimension reasons $E=\Endo(A)= \End_{\overline{k}}(A)
  \tensor \Q$, {\it i.e.\ }all $\overline{k}$-endomorphisms are
  already defined over $k$.  Note that the surjectivity in Theorem
  \ref{theorem}(1) ensures conversely that this case really does occur
  over any $k$!  Conversely, we know from the theory of supersingular
  elliptic curves that if $A$ is a supersingular elliptic curve over
  $k$ whose $\overline{k}$-endomorphisms are defined over $k$ then $E
  = {\rm{End}}_k^0(A)$ is a quaternion division algebra over $\Q$, so
  the central subfield $F = \Q(\pi) \subseteq E$ must be $\Q$.
 
\item When $a$ is odd, we have $F = \Q(\pi) \isom \Q(p^{1/2})$. In $F$
  we have two real places, and the invariant of $E$ at those two
  places is $1/2$ by the theorem. Since there is only one place $v$ of
  $F$ dividing $p$, this means that $E$ is not ramified at $v \mid p$.
  The division algebra $E$ has order 2 in $\Br(F)$, so $[E:F]=4$ by
  Theorem \ref{thm3.7}.  By formula~(\ref{eq:2}), $\dim A =2$. The
  characteristic polynomial $f_A$ of $\pi_A$ in $\Z[T]$ is therefore a
  quartic with $\pm p^{a/2}$ as a root.  By Theorem \ref{motivation},
  $f_A$ is a power of an irreducible polynomial in $\Q[T]$ since $A$
  is $k$-simple, so $f_A = (T^2-p^a)^2$.  Thus, the
  characteristic polynomial of the $q^2$-Frobenius for $A$ over a
  degree 2 extension $k'$ of $k$ is $(T-p^a)^4$.  By the existence
  aspect in the ``real case'' (1) applied over the field $k'$ with
  size $p^{2a}$, there exists a supersingular elliptic curve $A_0$
  over $k'$ with characteristic polynomial $(T-p^a)^2$.  By Theorem
  \ref{motivation} all choices of $A_0$ are $k'$-isogenous, and in (1)
  we saw that such $A_0$ are precisely the supersingular elliptic
  curves over $k'$ whose ``geometric'' endomorphisms are defined over
  $k'$.
  
  The characteristic polynomial of the $q^2$-Frobenius of $A_0\times A_0$
  is $(T-p^a)^4$, so $A_{/k'}$ is $k'$-isogenous to $A_0 \times A_0$.
  Thus, this is the case of an abelian surface that is $k$-simple but
  is isogenous over $k'$ to $A_0 \times A_0$ for some (equivalently, any)
  supersingular elliptic curve $A_0$ over $k'$ whose ``geometric''
  endomorphisms are all defined over $k'$.  Observe that necessarily
  $A$ is $k'$-isogenous to ${\rm{Res}}_{k'/k}(A_0)$.  By the
  surjectivity aspect of Theorem \ref{theorem}(1) such a case must
  always occur. (That is, this restriction of scalars must be
  $k$-simple for any finite field $k$ of size $p^a$ with odd $a$ and
  ``the'' quadratic extension $k'/k$.)
\end{enumerate}
\subsection{``General'' case: $F=\Q(\pi)$ is totally imaginary.}
For any embedding $\phi: \Q[\pi] \hookrightarrow \C$ we have $\phi(\pi)
\overline{\phi(\pi)} =q$, so $\overline{\phi(\pi)}= q/\phi(\pi)=\phi(q/\pi)$.
Thus, $\phi(F) \subseteq \C$ is stable under complex conjugation with
$\phi^{-1}(\overline{\phi(x)})$ independent of $\phi$ since this is true for
$x=\pi$. Then $F$ is a CM-field, {\it i.e.\ }a quadratic totally
imaginary extension of a totally real field $F_0$. Let $\overline{x} =
\phi^{-1}(\overline{\phi(x)})$ for any $x \in F$, so $\overline{\pi} =
q/\pi$.  Thus, $F_0 = \Q(\pi + \pibar)$ is the totally real subfield of
$F$. The division algebra $E$ splits at each place $v$ which does not
divide $p$. (There are no real places.)  For a place $v$ dividing $p$,
in $\Q/\Z$ we have
\begin{gather}\label{eq:3}
  \inv_v(E) + \inv_{\overline{v}}(E) = 0  \text{  , if } 
\overline{v} \neq v,
 \\\label{eq:4} \inv_v(E) =  0 \text{ , if }\overline{v} = v.
\end{gather}
Formula~(\ref{eq:3}) follows from $\pi \pibar = q$, $v(\pi) + v(\pibar)
= v(q)$, and the formula for the invariant in (\ref{eq:1}) since
$[F_{\overline{v}}:\Q_p] = [F_v:\Q_p]$.  Equation~(\ref{eq:4}) follows
from formula~(\ref{eq:1}) of the theorem and the fact that $[F_v
:\Q_p]$ is even when $v = \overline{v}$.

If $k$ is the prime field, $k=\F_p$, then $E$ is split everywhere
since $v(\pi)/v(p)$ is an integral multiple of 
$1/e_v$ and $e_v|[F_v:\Q_p]$ (so for any prime $v$ dividing $p$,
formula~(\ref{eq:1}) of the theorem shows that $\inv_v(E)=0$ in $\Q/
\Z$).  Thus, $E=F$ and hence $\End_k(A)$ is commutative (recall
$k = \F_p$).

\subsection{The remaining possibilities when $\dim A =1$} If $\dim A
=1$, then $2 = [E:F]^{1/2} [F:\Q]$. The case $F = \Q$ has been
settled, as has the case of real quadratic $F$ (with $A$ necessarily
an abelian surface).  Thus, if $A$ is either an ordinary elliptic
curve or a supersingular elliptic curve whose ``geometric''
endomorphism ring (a quaternion division algebra over $\Q$ ramified at
precisely $p$ and $\infty$) is strictly larger than $E$ then $E=F$ is an
imaginary quadratic field. By Theorem~\ref{theorem} this exhausts all
cases with $E=F$ and $F$ imaginary quadratic.  Since $F = \Q(\pi)$, we
conclude (via \S\ref{realcase}) that these are precisely the elliptic
curves over finite fields of size $q$ such that the $q$-Frobenius is
{\em not} multiplication by a rational integer.
\section{Injectivity of the map $A \mapsto \pi_A$}
\begin{proof}[Injectivity proof]
  Let $A$ and $B$ be two $k$-simple abelian varieties such that $\pi_A
  = \pi_B$ in $W(q)$ modulo conjugacy. Let $f_A$ and $f_B$ be the
  characteristic polynomials of $\pi_A$ and $\pi_B$, respectively. Since
  $A$ is simple, $f_A$ is a power of an irreducible polynomial $h \in
  \Q[T]$, say $f_A= h^{m}$ with $m \geq 1$. Since $\pi_A =\pi_B$ and $B$
  is simple, $f_B= h^n$ with $n \geq 1$. By relabeling we may assume $m
  \geq n$, so $f_B$ divides $f_A$.  Then by Theorem \ref{motivation} we
  see that $B$ is isogenous to an abelian subvariety of $A$ over $k$.
  Since $A$ and $B$ are $k$-simple, this inclusion must be an isogeny.
\end{proof}

\section{Surjectivity of the map $A \mapsto \pi_A$}
To prove surjectivity we have to construct sufficiently many simple
abelian varieties over $k$ to exhaust all the conjugacy classes of
Weil $q$-integers. But we cannot easily write down abelian varieties
of dimension greater than 2 over $k$, so how to proceed? We will
construct abelian varieties with complex multiplication over $\C$,
realize them over number fields that we
embed in finite extensions of $\Q_p$, and then reduce them to
obtain abelian varieties over finite fields.

{\bf Notation:} We will call a Weil $q$-integer $\pi$ {\em effective} if
$\pi$ is conjugate to the $q$-Frobenius of a simple abelian variety
over $k$. (That is, there exists a $k$-simple abelian variety $A$ and
an embedding $\Q[\pi] \hookrightarrow \End_k^0(A)$ carrying $\pi$ to $\pi_A$.)

So our goal is to prove that all Weil $q$-integers are effective.
In the definition of effectivity there is no loss of generality in
allowing $A$ to be $k$-isotypic (i.e., isogenous to a power of a $k$-simple
abelian variety) rather than $k$-simple, since such an $A$ has endomorphism
algebra that is a matrix algebra over a division algebra and so its central
Frobenius endomorphism $\phi$ generates a commutative central subfield that preserves
 any $k$-simple abelian subvariety and on which $\phi$ is its Frobenius endomorphism.  
 A basic example of an isotypic abelian variety over a field is a CM abelian variety
 (i.e., an abelian variety whose endomorphism algebra contains a commutative subfield with
 degree over $\Q$ equal to twice the dimension of the abelian variety). 
We will first prove the following lemma.
\begin{lemma}
Let $N$ be an integer, $N \geq 1$. If $\pi^N$ is effective, then $\pi$
is effective as well.
\end{lemma}
\begin{proof}
Let $k'$ be an extension of $k$ of degree $N$. Let $A'$ be a simple abelian
variety over $k'$ such that $\pi^N$ is conjugate to $\pi_{A'}$. Let $A$
be the abelian variety over $k$ which is the restriction of scalars of
$A'$. 

Let $\ell$ be a prime, $\ell \neq p$.
From the functorial characterization of Weil restriction of scalars, as
$\Q_{\ell}[G_k]$-modules (with $G_k=\Gal(\overline{k}/k)$ for a
separable closure $\overline{k}/k'$) we have
\[
V_{\ell}A = \Ind_{G_{k'}}^{G_k}V_{\ell}A'.
\]
From this one can see that the characteristic polynomial $f_A$ of
$\pi_A$ and the characteristic polynomial $f_{A'}$ of $\pi_{A'}$ are
related via $f_A(T)= f_{A'}(T^N)$ (Lemma~\ref{Brian1} in the Appendix).
If $\prod A_i$ is the isogeny decomposition of $A$ into $k$-simple factors, 
then $\pi$ is a root of $f_A = \prod f_{A_i}$, so $\pi$ is
a root of some $f_{A_i}$, say $i = 1$ without loss of generality.  Since $A_1$ is $k$-simple, $f_{A_1}$ is
a power of an irreducible polynomial $h$ over $\Q$  and 
the Frobenius endomorphism $\pi_{A_1}$ satisfies $h(\pi_{A_1}) = 0$ on $A_1$,
or more specifically the $\Q$-subalgebra
$\Q[\pi_{A_1}] \subseteq {\rm{End}}^0_k(A_1)$ is abstractly isomorphic to
$\Q[x]/(h(x))$ via $x \mapsto \pi_{A_1}$.   Thus, we can define
an embedding $\Q[\pi] \hookrightarrow {\rm{End}}^0_k(A_1)$ with
$\pi \mapsto \pi_{A_1}$.
\end{proof}
%\begin{definition}
%An algebraic number field $L$ is a {\em CM field} if $L$ is a quadratic
%totally imaginary extension of a totally real field $L_0$.
%\end{definition}
Let $\pi$ be a Weil $q$-integer.  In the following let $E$ be the
division algebra obtained from $\pi$ and the information in part (2) of
Theorem~\ref{theorem}; {\it i.e.}, $E$ is the central division algebra 
over $F=\Q(\pi)$ whose local invariants are as in the theorem and
so are completely determined by $\pi$. 
(Explicitly, ${\rm{inv}}_v(E) = 1/2$ for real places
$v$ of $F$, if any exist, and for non-archimedean
$v$ the fraction
$v(\pi)[F_v:\Q_p]/v(q)$ represents
${\rm{inv}}_v(E) \in \Q/\Z$; this vanishes
for such $v$ not over $p$ since $\pi$ is a Weil
$q$-integer.) By Theorem~\ref{exact}, such an $E$
indeed exists and is unique up to isomorphism.
\begin{lemma}
There exists a CM field $L$ containing $F=\Q(\pi)$ such that $L$
splits $E$ and such that $[L:F]=[E:F]^{1/2}$.
\end{lemma}
\begin{proof}
Since $\pi$ is a Weil $q$-integer,
$F$ is a totally real or CM field.  In the totally real
case, as in \S\ref{realcase} we must have $F = \Q$ or
$F = \Q(\sqrt{p})$ with $[E:F]^{1/2} = 2$, so
$L = F(\sqrt{-p})$ works.  If $F$ is CM with maximal
totally real subfield $F_0$, then we take
$L = F \otimes_{F_0} L_0$ with
$L_0/F_0$ a totally real extension of
degree $[E:F]^{1/2}$ such that for
all places $v_0$ of $F_0$ over $p$ we have two
properties:  (i) there
is exactly one place $w_0$ of $L_0$ over $v_0$,
and (ii) $w_0$ is unramified (resp. totally ramified)
over $v_0$ if the quadratic extension $F/F_0$ is
ramified (resp. unramified) over $v_0$.  Such an $L_0$
can be found by weak approximation, and by construction
for each place $w$ of $L$ over $p$ with
restrictions to $L_0$, $F$, and $F_0$ denoted
$w_0$, $v$, and $v_0$, we have
$$[L_w:F_v] = [L_{0,w_0}:F_{0,v_0}] = [L_0:F_0] = [E:F]^{1/2}.$$
Hence, $L \otimes_F E$ is everywhere locally split over $L$, so
it is split.
\end{proof}
\begin{definition}
  Let $A$ be an abelian variety over a field $K$. Let $L$ be a
  number field. We say that $A$ is {\em of type} ($L$) if there is a
  ring map $i: L \to \End_K^0(A)$ such that $[L:\Q] = 2 \dim A$.
\end{definition} 
\begin{lemma}\label{lemma3}
Let $L$ be a CM field containing $F=\Q(\pi)$ and such that $L$ splits
$E$. Then there exists an abelian variety $A$ of type $(L)$ over a
finite extension of $\Q_p$ such that $A$ has good reduction and such
that the Frobenius of its reduction is a power
$\pi^N$ of $\pi$.
\end{lemma}
Note that in this lemma there is no ``up to Galois conjugacy'' ambiguity;
we are saying that the entire CM field $L$ acts on $A$ (in the isogeny category
over a $p$-adic field) such that the element $\pi^N$ in its subfield $F$ is
the Frobenius endomorphism of the reduction of $A$.  
We will prove Lemma~\ref{lemma3} in the next subsection. The three lemmas
together clearly show that any given Weil $q$-integer $\pi$ is
effective.
\subsection{Proof of Lemma~\ref{lemma3}}
Let $L$ be a CM field with $[L:\Q]= 2 \dim A$. Let $\rho$ be the
automorphism of $L$ of order 2 which is induced by complex conjugation
for each embedding $L \hookrightarrow \C$. Intrinsically, $\rho$ is the
unique nontrivial automorphism of $L$ over its maximal totally real
subfield.  Let $C$ be an algebraically closed field of characteristic
0.  Let $\Phi$ be a CM type for $L$, {\it i.e.\ }$\Phi$ is a subset of
$\Hom_{\Q{\rm{-alg}}}(L,C)$ such that
\begin{gather}\label{CM:1}
\Phi \cap \Phi\rho = \emptyset \text{ and }\\
\label{CM:2}
\Phi \cup \Phi \rho = \Hom(L,C).
\end{gather}
\begin{definition}
  An abelian variety $A$ over a subfield $F \subseteq C$ is {\em of type
    $(L,\Phi)$}, if $A$ is of type $L$ over $F$ and the action of $C
  \tensor_{\Q} L$ on the tangent space $t_{A_C}$ is through the
  quotient $\prod_{\phi \in \Phi}C_{\phi}$ (where $L$ acts on $C_{\phi} =C$
  through $\phi$).
\end{definition}
\begin{lemma}\label{lemma4}
There exists an abelian scheme of type $(L,\Phi)$ defined over the
ring of integers of a number field contained in $C$.
\end{lemma}
\begin{proof}
This is proved in the notes for Tong Liu's talks in the seminar.
\end{proof}
From now on let us assume that $C$ is an algebraic closure of $\Q_p$.
For each place $w$ of $L$ such that $w \mid p$, let $L_w$ be the
completion of $L$ with respect to $w$.

We identify $\Hom_{\Q_p}(L_w,C)$ with its image in $\Hom(L,C)$ and
denote this image by $H_w$ (so $\Card(H_w) = [L_w:\Q_p]$). Let $\Phi_w:= \Phi\cap H_w$. We have a
decomposition
\begin{equation}
\Q_p\tensor L = \prod_{w  \mid  p} L_w
\end{equation}
and the disjoint unions
\begin{gather}
\Hom(L,C) = \bigcup_{w \mid p} H_w\\
\Phi= \bigcup_{w \mid p}\Phi_w.
\end{gather}
\begin{lemma}\label{lemma5}
  Let $A$ be an abelian scheme of type $(L,\Phi)$ defined over the ring
  of integers $\OO$ of a finite extension of $\Q_p$. Let $k_0$ be the
  residue field of $\OO$, and let $q_0=\Card(k_0)$. Let $A_0$ be the
  reduction of $A$ modulo the maximal ideal of $\OO$. Then there
  exists an element $\pi_0 \in L$ such that $i(\pi_0) \in \End(A)$ induces
  the Frobenius $\pi_{A_0} \in \Endo(A_0)$, and we have
\begin{equation}
\frac{w(\pi_0)}{w(q_0)}= \frac{\Card(\Phi_w)}{\Card(H_w)} = \frac{\Card(\Phi_w)}{[L_w:\Q_p]}
\end{equation}
for each $w \mid p$.
\end{lemma}
\begin{proof}
  This was proved by Shimura and Taniyama and will be presented in the
  seminar by Brian Conrad.
\end{proof}
Now we can prove Lemma~\ref{lemma3}.
\begin{proof}[Proof of Lemma~\ref{lemma3}]
Let $\pi,q,F,E$ as before, and assume that $L$ satisfies the
hypotheses of Lemma~\ref{lemma3}. Let $z \mapsto z^{\rho}$ be the
complex conjugation on $L$. We will now show how we can choose
the set $\Phi$ such that we have
\begin{equation}\label{eq:11}
\frac{w(\pi)}{w(q)}=\frac{\Card(\Phi_w)}{\Card(H_w)}\text{ for each
  place } w \text{ of }L\text{ with }w \mid p.
\end{equation}
Let $w$ be one such place, and let $v$ be the place of $F$ such that
$w \mid v$.
Let us put 
\[
n_w = \frac{w(\pi)}{w(q)}\cdot \Card(H_w) =
\frac{w(\pi)}{w(q)}\cdot[L_w:\Q_p] =
\frac{v(\pi)}{v(q)}\cdot[L_w:F_v]\cdot[F_v:\Q_p].\]

By functoriality of the local invariants and the hypothesis on
$\inv_v(E)$, $\inv_w(L \tensor_F E) =[L_w:F_v] \frac{v(\pi)}{v(q)}
[F_v:\Q_p] =n_w$ in $\Q / \Z$.  The fact that $L$ splits $E$ implies
that $n_w$ is an integer for each $w$.  We also clearly have
\begin{equation}\label{cond:1}
n_w\geq 0,
\end{equation}
and since $\pi \pi ^{\rho} = q$ we can conclude
\begin{equation}\label{cond:2}
n_{w} + n_{\rho w} = \Card H_w =\Card H_{\rho w}.
\end{equation}
We will now choose the CM type $\Phi$ by choosing suitable sets $\Phi_w$
and then letting $\Phi:= \bigcup_w \Phi_w$.  Given integers $n_w$
satisfying conditions~(\ref{cond:1}) and (\ref{cond:2}) we can see
that we can choose subsets $\Phi_w \subset H_w$ such that their union $\Phi:=
\bigcup_w\Phi_w$ satisfies conditions (\ref{CM:1}) and (\ref{CM:2}) and
$n_w = \Card \Phi_w$.  Let us make such a choice of $\Phi$. Then condition
(\ref{eq:11}), namely
\[\frac{w(\pi)}{w(q)} = \frac{\Card(\Phi_w)}{\Card(H_w)}\] is satisfied by 
construction.  By Lemma~\ref{lemma4} there exists an abelian scheme of
type $(L,\Phi)$ defined over the ring of integers $\OO$ of a finite
extension of $\Q_p$, and by Lemma~\ref{lemma5} there exists $\pi_0 \in
L$ such that
\begin{equation}
\frac{w(\pi)}{w(q)}=
\frac{\Card(\Phi_w)}{\Card(H_w)}=\frac{w(\pi_0)}{w(q_0)}.
\end{equation}
Here $\pi_0$ is a Weil $q_0$-integer, with $\pi_0 \in L$, which is
conjugate to the Frobenius of the reduction $A_0$ of $A$. Now we are
almost finished. If we replace the ring $\OO$ in Lemma~\ref{lemma5} by
an unramified extension of degree $N_0$ we replace $\pi_0$ by
$\pi_0^{N_0}$. We can now finish the proof of Lemma~\ref{lemma3} with
the following lemma.\end{proof}
\begin{lemma}\label{lemma6}
  Let $\pi$ be a Weil $q$-integer in $L$ and $\pi_0$ a Weil $q_0$-integer
  in $L$ which satisfy $\frac{w(\pi)}{w(q)}= \frac{w(\pi_0)}{w(q_0)}$
  for all $w$.
  Then there exist positive integers $N$ and $N_0$ such that $\pi^N =
  \pi_0^{N_0}$.
\end{lemma}
\begin{proof}
  By replacing $\pi$ and $\pi_0$ by powers we may assume that $q=q_0$.
  Then $w(\pi)=w(\pi_0)$ for each place $w$ of $L$ such that $w \mid p$.
  At the other finite places $\pi$ and $\pi_0$ are units, because they
  divide a power of $p$. At the infinite places $\pi$ and $\pi_0$ have
  the same absolute value, namely $q^{1/2}$. Hence $\pi/ \pi_0$ has
  absolute value 1 at each place of $L$, which implies that $\pi/ \pi_0$
  is a root of unity. Hence some power of $\pi / \pi _0$ is equal to 1,
  and the lemma follows.
\end{proof}
\section{Proof of part 2 of Theorem~\ref{theorem}}\label{sec8}
In this section we will prove the end of
Theorem~\ref{theorem}(1) and Theorem~\ref{theorem}(2). 

\subsection{Statement of result and preliminary arguments}

Let $A$ be a simple abelian variety over a finite field $k$ with size
$q$. Let $\pi_A$ be the $q$-Frobenius of $A$. Let $E:=\End^0_k(A)$.
Then $E$ is a division algebra. Let us recall Theorem~\ref{theorem}(2)
and the end of Theorem~\ref{theorem}(1):
\begin{theorem}
$\left.\right.$
\begin{enumerate}
\item \label{part:1}The center of $E$ is $F=\Q(\pi)$.
\item \label{part:2} $E$ splits at every finite place $v$ of $F$ such
  that $v$ does not divide $p$.
\item \label{part:3} $E$ does not split at any real place.
\item The invariant $\inv_v(E)$ for a place $v \mid p$ is
  given by the formula
\begin{equation}\label{formula:1}
\inv_v(E) = \frac{v(\pi _A )}{v(q)}[F_v:\Q_p] \bmod 1.
\end{equation}
\item \label{part:4}We have $2 \cdot \dim A = [E:F]^{1/2} [F:\Q]$.
\end{enumerate}
\end{theorem}
Formula~(\ref{formula:1}) will be proved separately. Everything else
will be proved now.
\begin{proof}
  Part~(\ref{part:1}): We know that $E$ is a division algebra, so its
  center is a field.  Let $G_k:=\Gal(\overline{k}/k)$. For a prime
  $\ell \neq p$ let $T_{\ell} A$ be the $\ell$-adic Tate module and let
  $V_{\ell}(A):= T_{\ell} A \tensor \Q_{\ell}$. Let $F_{\ell}:=
  \Q_{\ell} \tensor F$, and let $E_{\ell}:=E \tensor \Q_{\ell}$. Let
  $S:=
  \End{V_{\ell}}(A)$, and $T:=F_{\ell}$. From Tate's isogeny theorem
  we have $E \tensor \Q_{\ell} \isom \End_{G_k}(V_{\ell} A)$, so the
  centralizer of $T$ in $S$ is $E_{\ell}$ because $G_k$ is
  topologically generated by the $q$-Frobenius.  Since $E_{\ell}$ is
  semisimple, we can apply the double centralizer theorem and conclude
  that the centralizer of $E_{\ell}$ in $\End{V_{\ell}}(A)$ is
  $F_{\ell}$.  So $F_{\ell}$ is the center of $E_{\ell}$ and hence $F$
  is the center of $E$.\qed
  
  Part~(\ref{part:4}): If $A$ is a $k$-simple abelian variety, then
  the characteristic polynomial $f_A$ of $\pi_A$ is a power of a
  $\Q$-irreducible monic polynomial $h_A$.  Then $\deg{h_A}=[F:\Q]$. Say
  $f_A= {h_A}^e$. This implies that $2 \dim A =\deg {f_A} = e \cdot \deg
  h_A= e \cdot [F:\Q]$. We also have $[E:\Q]=e^2 \cdot [\Q(\pi _A):\Q]$. This
  comes from the dimension computation in the proof that
  $E\tensor_{\Q}\Q_{\ell} \isom \End_{G_k}(V_{\ell}A)$ in
  \cite{TateInv}.  (Alternatively, see
  \cite[Thm.~2(a),~Appendix~I]{Mumford} for a proof.)
  
  Part~(\ref{part:2}): Since $f_A=h_A^e$ with $h_A$ irreducible over
  $\Q$ and since $V_{\ell} A$ is a semisimple $F\tensor
  \Q_{\ell}$-module, for a prime $\ell$ different from $p$, the Tate
  module is a free module of rank $e$ over $F_{\ell}=F\tensor
  \Q_{\ell}$.  By Tate's theorem, we have $F_{\ell} \tensor _FE =
  \Q_{\ell} \tensor_\Q E = E_{\ell}$ is isomorphic to
  $\End_{F_{\ell}}V_{\ell}(A)$, that is, to the algebra of $e \times e$
  matrices over $F_{\ell}$. Since $F_{\ell}= \prod_{v \mid \ell} F_v$ it
  follows that $E_v=F_v\tensor_{\Q}E$ is a matrix algebra over $F_v$
  for $v\mid \ell$.  That means $E$ splits at all primes $v\mid \ell$.
 
  Part~(\ref{part:3}): Let $A$ be a simple abelian variety over $k$.
  Assume that $F=\Q(\pi)$ has a real place $v$. We may then view $\pi$
  as a real number, which has absolute value $q^{1/2}$, where
  $q=\Card(k)$. Consequently $\pi$ is a root of the polynomial $T^2
  -q$.
\begin{itemize}
\item If $q$ is a square, then $\pi$ is rational, $F=\Q$, and since we
  already proved that the formula $2 \dim A = [E:F]^{1/2}[F:\Q]$
  holds, we can rule out the possibility that $E=F=\Q$. Hence $E$ is
  non-split.  We also proved that $F$ is the center of $E$, and that
  $E$ splits at every finite place $v$ of $F$ that does not divide
  $p$.  By Theorem~\ref{exact}, $E$ must be non-split at $p$ and $\infty$,
  and the invariant must be $1/2$ at those two places.  This implies
  $[E:F]=4$, so $E$ is a quaternion division algebra which does not
  split at $p$ and the real place $v=\infty$. That means that the
  reasoning in \S\ref{realcase} applies, and so $A$ is a supersingular
  elliptic curve over $k$ whose $\overline{k}$-endomorphisms are all
  defined over $k$ (and necessarily $f_A = (T - q^{1/2})^2$).
\item If $q$ is not a square in $\Q$, then $F \isom \Q(\sqrt{p})$ is a
  real quadratic field. Also, since $f_A$ must be a power of an
  irreducible polynomial over $\Q$, we must have $f_A = (T^2 - q)^e$
  for some $e \geq 1$.  Over a quadratic extension $k'/k$, $A' =
  A_{/k'}$ therefore has associated $q^2$-Frobenius polynomial $f_{A'}
  = (T - q)^{2e}$, so by the case just treated we see that $A'$ is
  $k'$-isogenous to $A_0\times A_0$ for a supersingular elliptic curve
  $A_0$ over $k'$ whose ``geometric'' endomorphisms are all defined
  over $k'$.  By the previous case, the $k'$-endomorphism algebra of
  $A_0$ is the quaternion division algebra $D_p$ over $\Q$ that is
  ramified at exactly $p$ and $\infty$. Let $E'$ be the $k'$-endomorphism
  ring of $A'$. Then $E'$ is a $2 \times 2$ matrix algebra over $D_p$.
  Since $F=\Q[\pi]$, $E$ is the centralizer of $F$ in $E'$. By the
  remark after Theorem~\ref{Brian} in the appendix we have that $E \sim
  E'\tensor_{\Q} F \sim D_p \tensor_{\Q} F$ in $\Br(F)$.  Since $D_p$
  does not split over $\R$ it follows that $E$ does not split at
  either of the real primes of $F$.
\end{itemize}
\end{proof}

The rest of \S\ref{sec8} is devoted to the computation of $\inv_v(E)$
for the case $v\mid p$.  We will follow the method of Tate as explained
in \cite{MiWa}, except that we cut down on the input from
non-commutative ring theory.

\subsection{Dieudonn\'e modules}
To compute $\inv_v(E)$ for
$v \mid p$ we have to replace the $\ell$-adic Tate module by a
suitable object.

{\bf Notation:} Let $k$ be a perfect field of characteristic $p > 0$.
Let $W$ be the Witt vectors of $k$; {\it e.g.,} if $k$ is finite of
size $q = p^a$ then $W$ is the ring of integers in the unramified
extension of $\Q_p$ of degree $a$. Let $\sigma$ be the unique automorphism
of $W$ which reduces to the map $x \mapsto x^p$ on the residue field
$k$.  Let $\AAA:= W[\FF,\VV]$, where $\FF,\VV$ are indeterminates
subject to the relations
\begin{enumerate}
\item $\FF \VV = \VV \FF=p$
\item $\FF \alpha = \alpha^{\sigma} \FF$ and $\alpha \VV = \VV
  \alpha^{\sigma}$ for $\alpha \in W$.
\end{enumerate}
Elements of
the ``Dieudonn\'e ring'' $\AAA$ have
unique expressions as finite sums
$$a_0 + \sum_{j > 0} a_j \FF^j + \sum_{j > 0} b_j \VV^j$$
with
coefficients in $W$, so the center of $D_k$ is clearly $\Z_p[\FF^a,
\VV^a]$ if $k$ has finite size $p^a$ and it is $\Z_p$ otherwise
(i.e., if $k$ is infinite).  Some of the
main conclusions in classical Dieudonn\'e theory, as developed from
scratch in \cite[Ch.~I--III]{Fontaine}, are summarized in the
following theorem:
\begin{theorem}\label{Barsotti}
  There is an additive anti-equivalence of categories $G
  \rightsquigarrow \D(G)$ from the category of finite commutative
  $k$-group schemes of $p$-power order to the category of left
  $\AAA$-modules of finite length $($or equivalently, of finite
  $W$-length$)$.  Moreover,
\begin{enumerate}
\item A group
scheme $G$ has order $p^{\ell_W(\D(G))}$.
\item If $k \to k'$ is an extension of perfect fields with associated
  extension $W \to W'$ of Witt rings $($e.g., the absolute Frobenius
  automorphism of $k$$)$ then the functor $W' \otimes_W (\cdot)$ on Dieudonn\'e
  modules is naturally identified with the base-change functor on
  finite commutative group schemes.  In particular, $\D(G^{(p)})
  \simeq \sigma^{\ast}(\D(G))$ as $W$-modules.
\item Let  $F_G: G \to G^{(p)}$ be the relative Frobenius morphism.
The $\sigma$-semilinear 
action on $\D(G)$ induced by $\D(F_G)$ with respect
to the isomorphism $\D(G^{(p)}) \simeq \sigma^{\ast}(\D(G))$
equals the action of $\FF$,
and $G$ is connected if and only if $\FF$ is nilpotent on $\D(G)$. 
\end{enumerate}
\end{theorem}

Let $A$ be an abelian variety over $k$ with dimension $g \geq 1$. Let
$A[p^n]$ be the finite commutative $p^n$-torsion group scheme of rank
$(p^n)^{2g}$. We associate with $A$ the $p$-divisible group
$A[p^{\infty}]$ of height $2g$.  Rather generally, for any $p$-divisible
group $G = (G_n)_{n \geq 1}$ over $k$ with height $h \geq 1$, we let
$\D(G)$ denote the $\AAA$-module $\varprojlim \D(G_n)$. Then by the
same style of arguments used to work out the $\Z_{\ell}$-module
structure of Tate modules of abelian varieties in characteristic $\neq
\ell$ (resting on knowledge of the size of the $\ell$-power torsion
subgroups of geometric points), we use $W$-length to replace counting
to infer that $\D(G)$ is a free right $W$-module of rank $h$ with
\[
\D(G)/p^r\D(G)\stackrel{\sim}{\to}\D(G_r)
\] for all $r\geq 1$.  The $p$-divisible group $G$ is connected
if and only if $\FF$ is topologically nilpotent on $\D(G)$ (since this
is equivalent to nilpotence of $\FF$ on each $\D(G_r)$).

In analogy with the $\ell$-adic case we will now write $T_pG$ for
$\D(G)$ and $T_pA$ for the case $G = A[p^{\infty}]$. The $\AAA$-module
$T_pG$ will be the replacement for the $\ell$-adic Tate module in the
``classical'' case, even though it is contravariant in $A$; its
$\AAA$-action is the analogue of the Galois action on $\ell$-adic Tate
modules, though this Dieudonn\'e structure remains non-trivial when $k$ is
algebraically closed (whereas the Galois action on Tate
modules is trivial for such
$k$).

{\bf More Notation:} Let $L$ be the fraction field of $W$.  For any
$p$-divisible group $G$ over $k$, let $V_pG:=T_pG \tensor_{\Z_p} \Q_p$
(and write $V_pA$ for $G = A[p^{\infty}]$). Then $V_pG$ is an $L$-module
of rank equal to the height $h$ of $G$, and it also has a left module
structure over the ``Laurent polynomial ring'' $\AAA[1/p] =
L[\FF,1/\FF]$ that is non-commutative if $k \neq \F_p$.

Here is the analogue of Tate's isogeny theorem for $V_p A$:
\begin{theorem}\label{analogue}
For any abelian variety $A$ over $k$, 
with $p$-divisible group denoted
$A[p^{\infty}]$, the $\Q_p$-algebra map 
$$\Q_p \otimes_{\Q} {\rm{End}}_k^0(A) \to 
\End_k^0(A[p^{\infty}])$$ is an injection, and
if $k$ is finite then it is an isomorphism.
\end{theorem}
\begin{proof}
  By Theorem~\ref{Barsotti} this is $\Q_p \tensor_{\Z_p}(-)$ applied
  to the map
\[
\Z_p \tensor \End_k(A) \to \End_k(A[p^{\infty}])
\]
and this latter map is proved to be injective exactly as in the
$\ell$-adic case for $\ell \neq p$, by working with (contravariant)
Dieudonn\'e modules instead of the $\ell$-adic Tate modules.  As for
surjectivity for finite $k$, we first note
that the $q$-Frobenius endomorphism $\pi$ acting $W(k)$-linearly
on $\D(A[p^{\infty}])$ has the same characteristic polynomial
(in $\Z[T]$) as it does as a $\Z_{\ell}$-linear
endomorphism of the $\ell$-adic Tate module of $A$ for
any $\ell \ne p$; this is true more
generally for any element of $\End_k(A)$, and its
proof is given by using Dieudonn\'e modules in the role
of Tate modules in the proof of \cite[Thm.~4,~p.~180]{Mumford}. 
Hence, we may 
copy Tate's proof of surjectivity essentially
verbatim (say, as in Appendix 1 of \cite{Mumford}) by replacing
$\ell$-adic Tate modules with Dieudonn\'e modules throughout,
except that \cite[Lemma~6,~App.~1]{Mumford} only applies
to compute the $L$-dimension of
the centralizer of
the semisimple $L$-linear endomorphism
$\FF^a = V_p(\pi)$ acting on the $L$-vector space $V_p(A)$.
(That is, we can compute $\dim_L {\rm{End}}_{L[\FF^a]}(V_p(A))$.)
In order to conclude the proof,
it is necessary and sufficient to prove that this $L$-dimension
is equal to the $\Q_p$-dimension of
$\End_k^0(A[p^{\infty}]) = \End_{D_k[1/p]}(V_p(A))^{\rm{opp}}$.

The central subalgebra $F = \Q[\pi]$ in
the finite-dimensional semisimple $\Q$-algebra $\End_k^0(A)$ must
be semisimple and hence a finite product of fields;
we do not assume $A$ to be $k$-simple, so in particular
$F$ may not be a field.
Consider the decomposition of $A$ and $A[p^{\infty}]$
(in the isogeny sense) according to the idempotents of $F$.
The action by $F$ on 
$V_p(A)$ (through its action on $A$ in the isogeny
category of abelian varieties over $k$) 
commutes with the action by $D_k[1/p]$
on $V_p(A)$, and $\FF^a \in D_k[1/p]$ acts as $V_p(\pi)$ on $V_p(A)$.
Since there are no nonzero $k$-maps among the factor abelian varieties
or $p$-divisible groups over $k$ on which the central subalgebra $F$ acts
through distinct quotient fields, we may reduce the proof of
Tate's theorem to the case when $F$ is a field.

Let $m \in \Z[T]$ be the common characteristic
polynomial for $V_{\ell}(\pi)$ on the $\Q_{\ell}$-vector spaces $V_{\ell}(A)$
(for all $\ell \ne p$) and for $V_p(\pi)$ on the
$L$-vector space $V_p(A)$, so by the faithfulness of
$V_p(A)$ as a $\Q_p \otimes_{\Q} F$-module we see that 
${\rm{rad}}(m)$ is the minimal polynomial of
$\pi \in F$ over $\Q$.   Hence, 
$\Q_p \otimes_{\Q} F \simeq \prod_{v|p} F_v$ with $F_v \simeq
\Q_p[T]/(m_v)$ for the monic irreducible factors 
$m_v \in \Z_p[T]$ of $m$ over $\Q_p$.  Each $m_v(0)$
divides $m(0) = q^{\dim A}$, so all $m_v$'s have nonzero
constant term.  Since
$\Q_p \otimes_{\Q} F$ acts $D_k[1/p]$-linearly on $V_p(A)$,
we get a decomposition of $D_k[1/p]$-modules
$V_p(A) \simeq \prod_{v|p} V_p(G_v)$
where $\prod_{v|p} G_v$ is the isogeny decomposition
of $A[p^{\infty}]$ with respect to the idempotents
of $\Q_p \otimes_{\Q} F$.   Since
the central element
$m_v(\FF^a) \in D_k$ acts on $G_v$ through
the element $m_v(\pi) = 0$ in $F_v$, $V_p(G_v)$ is a left module over
the quotient algebra $C_v = D_k[1/p]/D_k[1/p]m_v(\FF^a)$.
Using the compatible decompositions (as $L$-algebras and $\Q_p$-algebras
respectively)
$$\End_{L[\FF^a]}(V_p(A)) \simeq \prod_{v|p}
\End_{L[\FF^a]}(V_p(G_v))$$
and
$$\End_{D_k[1/p]}(V_p(A)) \simeq
\prod_{v|p} \End_{C_v}(V_p(G_v)),$$
we are reduced to proving 
$$\dim_L \End_{L[\FF^a]}(M_v) \stackrel{?}{=}
\dim_{\Q_p} \End_{C_v}(M_v)$$
for any $v|p$ and any left $C_v$-module $M_v$ with finite $\Q_p$-dimension
(e.g., $M_v = V_p(G_v)$).  This general equality
is valid when $C_v$ is replaced
with $D_k[1/p]/D_k[1/p]h(\FF^a)$ for
any $h \in \Q_p[T]$ that is
a monic irreducible with nonzero
constant term; a proof will be
given in Corollary \ref{newcor} as a consequence of some
general calculations in non-commutative algebra
(having no logical dependence on abelian varieties
or $p$-divisible groups).
\end{proof}

\subsection{Restatement via left $D_k[1/p]$-modules}\label{restate}

Take $k$ to be finite of size $q$, and $A$ a $k$-simple
abelian variety over $k$ with endomorphism algebra $E$, so $E$ is a
central simple algebra over 
the subfield $F = \Q[\pi]$.  Under the decomposition $E
\otimes \Q_p \simeq \prod_{v|p} E_v$ with $E_v = E \otimes_F F_v$, we get a
corresponding decomposition of $A[p^{\infty}]$ in the isogeny category of
$p$-divisible groups over $k$, as a product of {\em nonzero}
$p$-divisible groups $G_v$ on which $F \otimes \Q_p$ acts through its
quotients $F_v$ (here we use the elementary injectivity aspect of
Theorem \ref{analogue} to know that each $G_v$ is nonzero).

Since the Dieudonn\'e functor is fully faithful and contravariant, and
the map in Theorem \ref{analogue} is an isomorphism for finite $k$, we
may identify the central $F \otimes_{\Q} \Q_p$-algebra $E \otimes \Q_p$ with the
opposite $F \otimes_{\Q} \Q_p$-algebra to ${\rm{End}}_{\AAA[1/p]}(V_p A)$,
where $F = \Q(\pi)$ acts through functoriality via its action on $A$
in the isogeny category over $k$.
In particular, $V_p(\pi)$ is the action of the central element $\FF^a
\in \AAA[1/p]$, and so we get $$E_v \simeq
\End_k^0(G_v) \simeq \End_{\AAA[1/p]}(V_p G_v)^{\rm{opp}}.$$
as finite-dimensional $F_v$-algebras.  We conclude that 
the right side is a central simple $F_v$-algebra, and our
problem is to compute its invariant.

\subsection{Passage to a central simple $F_v$-algebra
quotient of $D_k[1/p]$}

We continue with the notation and hypotheses as in \S\ref{restate}.
Since $\pi$ in the ring of integers of $F$ acts
on $A$ as the $q$-Frobenius
over $k$, on the finite-dimensional $\Q_p$-vector space $V_p(G_v)$ the
$\Q_p$-linear operator $\FF^a = V_p(\pi)$ acts with a characteristic
polynomial that divides the common characteristic polynomial $m \in
\Q[T] \subseteq \Q_p[T]$ for the $q$-Frobenius on each $V_{\ell}(A)$ over
$\Q_{\ell}$ (for $\ell \neq p$) and $V_p(A)$ over $L$.  (Explicitly, for
all $n \in \Z$ the integer $m(n)$ is the degree of $n - \pi$ acting on
$A$.) As we noted in the proof of Theorem \ref{analogue},
$m$ is a power of the minimal polynomial of $\pi
\in F$ over $\Q$, and so the decomposition $F \otimes_{\Q} \Q_p \simeq
\prod_{v|p} F_v$ corresponds to the pairwise distinct monic irreducible
factors $m_v$ of $m$ over $\Q_p$.  That is, $F_v = \Q_p(\pi)$ with $\pi
\in F_v$ having minimal polynomial $m_v$ over $\Q_p$.  In particular,
$m_v(0)$ divides $m(0) = q^{{\rm{dim}} A}$, so all $m_v$'s have
nonzero constant term.

For each $v|p$, the $F_v$-action on the nonzero $V_p(G_v)$ commutes with the
$D_k[1/p]$-action (as it arises from an action of $F_v$ on the
$p$-divisible group $G_v$ in the isogeny category over $k$), and the
central element $m_v(\FF^a) \in D_k[1/p]$ acts as multiplication by the
element $m_v(\pi) \in F_v$ that is zero.  In other words, $V_p(G_v)$ is
a nonzero module over the ring
$$C_v := D_k[1/p]/D_k[1/p]m_v(\FF^a).$$
Recall that $q = p^a$.

\begin{theorem}\label{csar}
Let $h \in \Q_p[T]$ be a monic irreducible
with $h \ne T$, and let
$K = \Q_p[T]/(h)$. The central $K$-algebra
$C = D_k[1/p]/D_k[1/p]h(\FF^a)$ $($with $T$ acting
as $\FF^a$$)$ is central simple.
\end{theorem}

In order to prove Theorem \ref{csar},
we require some preparations.  
Recall that $L$ denotes the fraction field of $W(k)$.
We have $\dim_L C = \deg(h(T^a)) = a [K:\Q_p]$
due to:

\begin{lemma}\label{divalg}
Let $\lambda \in L[T]$ be monic 
with positive degree $d$
and non-vanishing constant term.  Every $g \in D_k[1/p]$ can be uniquely
written 
$$g = h \cdot \lambda(\FF) + (c_0 + c_1 \FF + \cdots + 
c_{d-1}\FF^{d-1})$$
with $c_0, \dots, c_{d-1} \in L$ and $h \in D_k[1/p]$.
In particular, 
the left $D_k[1/p]$-module $D_k[1/p]/D_k[1/p] 
\cdot \lambda(\FF)$ has dimension $d$
as a left $L$-vector space. 
\end{lemma}

\begin{proof}
  The uniqueness is easily proved by chasing ``least-degree''
  coefficients (when things are nonzero).  For existence it suffices
  to treat the cases $g = \FF^r$ with $r \geq d$ and $r < 0$.  These go
  as in the commutative case (for $r < 0$ we use the non-vanishing of
  the constant term of $\lambda$).
\end{proof}


Since $C$ is an $L \otimes_{\Q_p} K$-module with finite
$\Q_p$-dimension, we obtain $\dim_{K} C = a[L:\Q_p]$.  Let $(L
\otimes_{\Q_p} K)[\FF]$ denote the $K$-algebra defined by the relations
$\FF^a = 1 \otimes \pi$ (for $\pi \in K$) 
and $\FF \cdot x = (\sigma \otimes 1)(x) \cdot
\FF$ for $x \in L \otimes_{\Q_p} K$, with $\sigma \in {\rm{Gal}}(L/\Q_p)$ the
absolute Frobenius.  This is easily checked to be a central
$K$-algebra with dimension $a[L:\Q_p] = a^2$.  There is a unique map of
central $K$-algebras
$$(L \otimes_{\Q_p} K)[\FF] \to C$$
sending $\FF$ to $\FF$ and $L$ to $L$ 
by the identity map, and this is clearly surjective.  By
consideration of $K$-dimensions, it is an isomorphism.  Theorem
\ref{csar} therefore will follow from:

\begin{theorem}\label{cyclicthm} The central $K$-algebra
$(L \otimes_{\Q_p} K)[\FF]$ is $K$-isomorphic
to a matrix algebra over a cyclic $K$-algebra.
\end{theorem}

\begin{rem} Cyclic algebras are a special class
  of central simple algebras, studied by Dickson before the advent of
  the general theory.
\end{rem}

\begin{proof} 
  We first construct the cyclic $K$-algebra $\Delta$ over which the
  $K$-algebra $(L \otimes_{\Q_p} K)[\FF]$ will be proved to be a matrix
  algebra.  Let $f$ be the absolute residue degree of $K$, and let
  $g := \gcd(f,a)$. 
  Let $L \cap K \subseteq K$ denote the maximal
  unramified subextension of $K$ that embeds into $L$ (over $\Q_p$),
  so $g=[L \cap K:\Q_p]$.  We arbitrarily choose one of the $g$
  embeddings $L \cap K \to L$ as cyclic extensions of $\Q_p$, and let
  $LK$ denote the resulting linearly disjoint compositum $L \otimes_{L \cap
   K} K$.
  
  Clearly $LK/K$ is unramified with degree $a/g$.  Let
$$\theta:L \otimes_{\Q_p} K \to LK$$
be the resulting projection map. Let
$\Delta$ be the central
$K$-algebra $(LK)[\FF']$ defined with relations
$(\FF')^{a/g}=\pi \in K$ and $\FF' \cdot x' = \sigma'(x') \cdot \FF'$ where
$\sigma':LK \simeq LK$ is the $L \cap K$-automorphism of order $a/g$
induced by $\sigma^g$ on $L$ and the identity 
on $K$.  Thus, $\sigma'(\theta(x))
= \theta((\sigma^g \otimes 1)(x))$ for 
$x \in L \otimes_{\Q_p} K$, and we have a
natural isomorphism of $K$-algebras
$$L \otimes_{\Q_p} K \simeq \prod_{j=1}^g LK$$
defined by
$$x \mapsto (\theta(x), \theta((\sigma \otimes 1)(x)), \dots,
\theta((\sigma^{g-1}\otimes 1)(x))).$$
The central
$K$-algebra $\Delta$ is a cyclic $K$-algebra; in classical notation,
$\Delta=(LK/K, \sigma',\pi)$, where $LK$ is a cyclic (unramified)
extension of $K$ and $\sigma'$ is a chosen generator
of its Galois group.

Now consider the natural map of central $K$-algebras
\[\xi:(L\tensor_{\Q_p} K)[\FF] \to M_g (\Delta)\]
defined by
\[ x \mapsto\left( \begin{array}{cccc}
\theta(x) & 0 & \dots& 0 \\
0 & \theta((\sigma \otimes 1)(x)) & 0&0 \\
\vdots & \vdots & \ddots &\vdots\\
 0  & 0 & \dots  &\theta((\sigma^{g-1} \otimes 1)(x))\end{array} \right)\] 
for $x \in L \otimes_{\Q_p} K$ and 
\[ \FF \mapsto\left( \begin{array}{ccccc}
0 & 1 & 0& \dots & 0 \\
0 & 0 & 1 &  \dots & 0 \\
\vdots& 0& \dots &\ddots& \vdots\\
\vdots & \vdots & 0 &\dots &1\\
 \FF'  & 0 & 0 &\dots & 0\end{array} \right)\] 
The map $\xi$ is obviously a well-defined $K$-algebra homomorphism,
and it is easy to check that both sides
have the same $\Q_p$-dimension.  To prove that
$\xi$ is an isomorphism, it suffices to check
surjectivity.  

The decomposition of $L \otimes_{\Q_p} K$ as a product of copies of
$LK$ shows that the image of $\xi$ contains the diagonal matrices
whose entries are elements of $LK \subseteq \Delta$.  
Moreover, $\xi(\FF^g)$
is the diagonal matrix whose diagonal entries are all equal to $\FF'
\in \Delta$.  Thus, by the definition of $\Delta$ we see that the image of
$\xi$ contains all diagonal matrices in $M_g (\Delta)$.
Left-multiplication by $\xi(\FF)$ carries the set of diagonal
matrices onto the set of matrices of the form
$$\left( \begin{array}{ccccc}
0 & \delta_2 & 0& \dots & 0 \\
0 & 0 & \delta_3 &  \dots & 0 \\
\vdots& 0& \dots &\ddots& \vdots\\
\vdots & \vdots & 0 &\dots &\delta_g\\
 \FF'\delta_1  & 0 & 0 &\dots & 0\end{array} \right)$$
with arbitrary $\delta_1, \dots, \delta_g \in \Delta$, and so
since $\FF' \in \Delta$ has a multiplicative inverse
(as $(\FF')^{a/g} = \pi \in K^{\times}$)
it follows that the image of $\xi$ contains the
matrix
$$\left( \begin{array}{ccccc}
0 & 1 & 0& \dots & 0 \\
0 & 0 & 1 &  \dots & 0 \\
\vdots& 0& \dots &\ddots& \vdots\\
\vdots & \vdots & 0 &\dots &1\\
1 & 0 & 0 &\dots & 0\end{array} \right).$$
Powers of this matrix, together with arbitrary
diagonal matrices in $M_g(\Delta)$,
generate $M_g(\Delta)$ under
the operations of multiplication and addition.
This establishes surjectivity.
\end{proof}

\begin{rem}
For later purposes, it is convenient to describe
the cyclic algebra $\Delta$ in terms of an arithmetic
Frobenius generator $\phi$ of the Galois group of the unramified
extension $LK/K$.  With notation as used above,
we claim that $\Delta = (LK/K, \sigma', \pi)$ is
$K$-isomorphic to the cyclic $K$-algebra
$(LK/K, \phi, \pi^{f/g})$.   This amounts to checking
that ${\sigma'}^{f/g} = \phi$ in ${\rm{Gal}}(LK/K)$,
and so we just have to compare these on the residue field
of $L$ (viewed inside that of $LK$).  Since $\sigma'$
reduces to $x \mapsto x^{p^g}$ on the residue field of $L$,
${\sigma'}^{f/g}$ reduces to $x \mapsto x^{p^f}$ on this
residue field.  But $f$ is the residual degree for $K$ over $\Q_p$, 
so we are done.
\end{rem}

\medskip

It is an elementary result in the theory of simple rings that (up to
isomorphism) there is a unique simple module over such a ring and that
all finite modules are isomorphic to a direct sum of copies of this
simple module.  Thus, up to isomorphism there is a unique simple
$D_k[1/p]$-module $V$ killed by the central element $h(\FF^a)$,
and so for any nonzero left $C$-module $M$ with
finite $\Q_p$-dimension we have $M \simeq V^{\oplus r}$ 
for some $r \geq 1$ (with $C$ denoting the quotient
of $D_k[1/p]$ modulo the central element
$h(\FF^a)$).  In particular,
$\End_{D_k[1/p]}(M)$ is a matrix algebra over the
division ring $D = \End_{C}(V)$.
Since there is an isomorphism of left $C$-modules
$C \simeq V^{\oplus r_0}$ for some $r_0 \ge 1$, 
so $M_{r_0}(D) = {\rm{End}}_C(C) = C^{\rm{opp}}$,
$D$ has center $K$ and in the Brauer group of $K$
we have $D \sim C^{\rm{opp}}$.
Hence, the class of $\End_{D_k[1/p]}(M)^{\rm{opp}}$
in ${\rm{Br}}(K)$ is the same as that of $C$
for any nonzero finitely generated left $C$-module $M$.
For any such $M$ we have the following result
that completes the final step in the proof of Theorem \ref{analogue}:

\begin{cor}\label{newcor} Let $h \in \Q_p[T]$ be a monic irreducible with
$h \ne T$, and let $C = D_k[1/p]/D_k[1/p]h(\FF^a)$.
For any finite left $C$-module $M$,
\begin{equation}\label{dimid}
\dim_L \End_{L[\FF^a]}(M) = \dim_{\Q_p} \End_{C}(M).
\end{equation}
\end{cor} 

Since $L[\FF,1/\FF] = D_k[1/p]$
and $\End_C(M) = \End_{L[\FF]}(M)$
($\FF$ acts invertibly on $M$, as $h(0) \in \Q_p^{\times}$), 
the equality $[L:\Q_p] = a$ suggests the possibility that one may be
able to deduce (\ref{dimid})
by some general nonsense with Galois descent.
However, it seems that the proof requires fine structural information
concerning $C$.

\begin{proof}
Let $V$ be a simple left $C$-module and
let $K = \Q_p[T]/(h)$.  We have
$M \simeq V^{\oplus r}$ for some $r \ge 0$. 
The two sides of (\ref{dimid}) for $M$ are each
$r^2$ times the two sides of (\ref{dimid}) for $V$
in the role of $M$.  Hence, the truth of (\ref{dimid})
is independent of the choice of $r$, so it suffices
to verify this identity for one $M \ne 0$.
We choose $M = C$.  Since $\End_C(C) = C^{\rm{opp}}$
(with action through right multiplications on $C$), 
$$\dim_{\Q_p} \End_C(C) = [L:\Q_p]\dim_L(C^{\rm{opp}}) = 
[L:\Q_p] \deg h(T^a)$$
by Lemma \ref{divalg} (with $\lambda = h(T^a)$ satisfying
$\lambda(0) \ne 0$ since the irreducible monic $h$ is not $T$).
Because $\deg h = [K:\Q_p]$ and $[L:\Q_p] = a$ we may rewrite this as
$$\dim_{\Q_p} \End_C(C) = \frac{[L \otimes_{\Q_p} K:\Q_p]a^2}{[L:\Q_p]},$$
and so by expressing an $L$-dimension as
$[L:\Q_p]^{-1}$ times a $\Q_p$-dimension it is equivalent to prove
$$\dim_{\Q_p} \End_{L[\FF^a]}(C) = [L \otimes_{\Q_p} K:\Q_p]a^2.$$
It therefore suffices to show that $\End_{L[\FF^a]}(C)$ is
isomorphic as a $\Q_p$-algebra to an $a \times a$ matrix
algebra over $L \otimes_{\Q_p} K$. 

Let $t \in K = \Q_p[T]/(h)$ denote
the residue class of $T$, so $K = \Q_p[t]$.  
We saw above Theorem \ref{cyclicthm} that 
as $L[\FF^a]$-algebras
$C \simeq (L \otimes_{\Q_p} K)[\FF]$
where the right side has relations
$\FF^a = 1 \otimes t$ and $\FF \cdot x = (\sigma \otimes 1)(x)$
for all $x \in L \otimes_{\Q_p} K$.  Hence, 
the $L[\FF^a]$-endomorphism ring of $C$
is the same as the $L \otimes_{\Q_p} K$-endomorphism ring of
$C$, and this in turn is indeed an $a \times a$ matrix
algebra over $L \otimes_{\Q_p} K$.
\end{proof}

\subsection{Computing with a cyclic algebra}

We now return to our original problem of
computing ${\rm{inv}}_v(E_v)$ where
$E_v = F_v \otimes_F \End_k^0(A)$
for a $k$-simple abelian variety $A$
and $F = \Q(\pi)$ is the central subfield of
$\End_k^0(A)$ generated by the $q$-Frobenius endomorphism.
By Theorem \ref{csar} (with $h = m_v$ and $K = F_v$),
$C_v = D_k[1/p]/D_k[1/p]m_v(\FF^a)$
is a central simple $F_v$-algebra.
Writing $V_v$ to denote a simple left $C_v$-module
(unique up to isomorphism) we have 
$V_p(G_v) \simeq V_v^{\oplus r_v}$ as left $D_k[1/p]$-modules
for some $r_v \ge 1$, so
the $F_v$-algebra $\End_{D_k[1/p]}(V_p(G_v))$ is a matrix
algebra over the division ring $D_v^{\rm{opp}} =
\End_{D_k[1/p]}(V_v)$.  For any $N \geq 1$, the $F_v$-algebra
$\End_{D_k[1/p]}(V_v^{\oplus N})$ is a matrix algebra over the division ring
$\End_{D_k[1/p]}(V_v)$ (as $F_v$-algebras), and so its class in 
${\rm{Br}}(F_v)$ is independent of $N$ (and is opposite the class
of $E_v$).  But $C_v$ is $D_k[1/p]$-isomorphic to a finite direct
sum of copies of $V_v$ as left $D_k[1/p]$-modules, so the
$F_v$-algebra $\End_{D_k[1/p]}(C_v)$ also lies in this same Brauer
class.  This endomorphism ring is the central simple $F_v$-algebra
$C_v^{\rm{opp}}$ via $C_v$ acting on itself through right
multiplications.  Thus, in ${\rm{Br}}(F_v)$,
$$E_v \simeq \End_{D_k[1/p]}(V_p(G_v))^{\rm{opp}} \sim
\End_{D_k[1/p]}(C_v)^{\rm{opp}} =
(C_v^{\rm{opp}})^{\rm{opp}} \simeq
C_v.$$
By the Remark following the proof of Theorem \ref{cyclicthm}
(with $K = F_v$)
this class is represented by
the cyclic $F_v$-algebra $$\Delta_v = 
(LF_v/F_v, \phi, \pi^{f_v/g_v})$$
that rests on the generator
$\phi$ of ${\rm{Gal}}(LF_v/F_v)$ and the
element $\pi^{f_v/g_v}$ in $F_v^{\times}$
(where $f_v = f(v|p)$ and $g_v = {\rm{gcd}}(f_v,a)$).
Since $[LF_v:F_v]=a/g_v$, by
the formula in Theorem \ref{localinv} for local invariants
of certain cyclic algebras over 
non-archimedean local fields we get the formula
\[
\inv_v(\Delta_v) = \frac{1}{a/g_v}\cdot v(\pi^{f_v/g_v})=(f_v/g_v) 
\cdot v(\pi) g_v/a = (f_v/a) \cdot
v(\pi).
\]
Let $e_v$ be the ramification index of $F_v$ over $\Q_p$. Then
$e_vf_v=[F_v:\Q_p].$ We have
\[
v(q) = a \cdot v(p) = a\cdot e_v,
\]
so $a= v(q)/e_v$. This implies that
\[
\inv_v(\Delta_v) = 
\frac{f_v}{a} \cdot v(\pi)= \frac{e_v f_v}{v(q)} \cdot v(\pi) =
\frac{v(\pi)}{v(q)} \cdot [F_v:\Q_p].
\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Appendix}

In this appendix we treat two topics: the interaction
of Frobenius endomorphisms with Weil restriction for
abelian varieties, and the formula for the local
invariant of certain cyclic algebras over a local field.

  Let $k'/k$ be an extension of finite fields with respective
sizes $q'$ and $q$. Let $A'$ be an abelian
  variety over $k'$, and $A = {\rm{Res}}_{k'/k}(A)$
its Weil restriction to $k$ (an abelian variety of
dimension $[k':k]\dim A$ over $k$). 
Let $R'\subseteq \End_{k'}^{0}(A')$, $R\subseteq
  \End_k^0(\Res_{k'/k} A')$ be the $\Q$-subalgebras generated by the
  $q'$-Frobenius $\pi'$ and the $q$-Frobenius $\pi$
  for $A'$ and $A$ respectively. 
\begin{lemma}\label{Brian1}
Via $\Res_{k'/k}: \End_{k'}^0(A') \to
  \End_{k}^0(\Res_{k'/k}A')$ we have $\pi' \mapsto \pi^{[k':k]}$. That is,
$\Res_{k'/k}(\pi')=\pi^{[k':k]}$.
\end{lemma}
\begin{proof}
  On the $\ell$-adic Tate modules, this says $\Ind_{k'}^{k}(\pi')$ on
  $\Ind_{k'}^{k}(V_{\ell} A')$ is $\pi^{[k':k]}$. This is clear since
  $G_k,G_{k'}$ are commutative and 
\[
\left( \begin{array}{ccccc}
0 & 0 &  \dots&0& \pi' \\
1 & 0& \dots& 0&0 \\
0& 1&\ddots &\vdots &\vdots\\
\vdots & \vdots &\ddots& \ddots &0\\
 0  & \dots &0 &1  &0\end{array} \right)^{[k':k]}=
\left( \begin{array}{ccc}
\pi' & \dots  &0 \\
0& \ddots &\vdots\\
 0  & \dots  &\pi'\end{array} \right)
\] 
as block matrices.
\end{proof}
Using the lemma, we have a map of $\Q$-algebras
\[\phi: R\tensor_{R'}\End_{k'}^0 (A') \to \End_{k}^0(\Res_{k'/k}A'),\]
 where $R'\to R$ via $\pi'\mapsto
  \pi^{[k':k]}$. (Since $R$ is in the center of the target.) 
\begin{theorem}\label{Brian}
  The map $\phi$ is an isomorphism.
\end{theorem}
\begin{proof}
Apply $\Q_{\ell} \tensor_\Q (-)$ to convert it into the natural map
\[
\Q_{\ell}[\pi]
\tensor_{\Q_{\ell}[\pi']}\End_{\Q_{\ell}[\pi']}(V_{\ell}A')\to
\End_{\Q_{\ell}[\pi]}(\Ind_{G_{k'}}^{G_k}V_{\ell}A').
\]
We view $V_{\ell}A'$ as a $\Q_{\ell}[T']$-module with $T'$ acting as
$\pi'$. We have \begin{eqnarray*}\Ind_{G_{k'}}^{G_k}T_{\ell}A' &\isom&
\Ind_{\Z_{\ell}[T',T'^{-1}]}^{\Z_{\ell}[T,T^{-1}]}T_{\ell}A'\\
 &=& \Z_{\ell}[T,T^{-1}] \tensor_{\Z_{\ell}[T',T'^{-1}]} T_{\ell}A'\\
&=&\Z_{\ell}[T]\tensor_{\Z_{\ell}[T']}T_{\ell}A'.
\end{eqnarray*}
Thus, $\Ind_{G_k}^{G_{k'}}V_{\ell}A' \isom \Q_{\ell}[T]\tensor
_{\Q_{\ell}[T']} V_{\ell}A'$. We can apply $\Qbar_{\ell}
\tensor_{\Q_{\ell}}(-)$, so we want to study the natural
$\Qbar_{\ell}[T]$- algebra map
\[
\Qbar_{\ell} [\pi] \tensor_{\Qbar_{\ell}[\pi']} \End_{\Qbar_{\ell}[T']}(M')
\to
\End_{\Qbar_{\ell}[T]}(\Qbar_{\ell}[T]\tensor_{\Qbar_{\ell}[T']} M') \,\,\,
\] 
for $\Qbar_{\ell}[T']$-modules $M'$ with $\pi'$ acting as $T'$ and
$\pi \mapsto T$. Note that $M'=V_{\ell} A'$ is a semisimple
$\Qbar_{\ell}[T']$-module on which $T'=\pi'$ invertibly.  Consider any
semisimple $\Qbar_{\ell}[T']$-module $M'$ on which $T'$ acts
invertibly. Since $\Qbar_{\ell}$ is algebraically closed, we have
\[ M' \isom \prod_{i=1}^{r}(\Qbar_{\ell}[T']/(T'-\alpha_i))^{\oplus
  e_i}\] as $\Qbar_{\ell}[T']$-modules for distinct $\alpha_1,\dots,
\alpha_r \in \Qbar_{\ell}^{\times}$.  Thus,
\[
\Qbar_{\ell}[T]\tensor_{\Qbar_{\ell}[T']} M' \isom
\prod_{i=1}^{r}(\Qbar_{\ell}[T]/(T^f-\alpha_i))^{\oplus e_i}
\]
as  $\Qbar_{\ell}[T]$-modules with $f=[k':k]$. We have, via the
Chinese Remainder Theorem and since all $\alpha_i \neq 0$,
$\Qbar_{\ell}[\pi]\stackrel{\sim}{\to}\prod
\Qbar_{\ell}[T]/(T^f-\alpha_i)$
as $\Qbar_{\ell}[\pi]$ is the $\Qbar_{\ell}$-subalgebra generated by
the $T$-action on $\Qbar[T]\tensor_{\Qbar_{\ell}[T']}M'$.

Thus, the map of interest is the natural map
\begin{gather*}
\prod_{i}(\Qbar_{\ell}[T]/(T^f-\alpha_i))\tensor_{\Qbar_{\ell}[T']}M_{e_i}(\Qbar_{\ell}[T']/(T'-\alpha_i)){\to}
  \\\prod_iM_{e_i}( \Qbar_{\ell}[T]/(T^f-\alpha_i)).
\end{gather*}
Since
\[
\Qbar_{\ell}[T]/(T^f-\alpha_i) \tensor_{\Qbar_{\ell}[T']}
\Qbar_{\ell}[T']/(T'-\alpha_i) \to \Qbar_{\ell}[T]/(T^f-\alpha_i)
\]
is an isomorphism, we are done.
\end{proof}
\begin{rem}
  In the special case $A'=k'\tensor_k A$, with $n = [k':k]$,
the $q$-Frobenius polynomial for $\Res_{k'/k}A'$ is the polynomial
$$\prod_{\zeta^n = 1} f_A(\zeta T) \in \Q[T]$$
with degree $n \deg f_A = 2n {\rm{dim}}A$, 
so if $f_A$ is a polynomial in $T^n$ then
this polynomial is $f_A^n = f_{A^n}$.  Hence,
assuming $f_A$ is a polynomial in $T^n$
we obtain that $\Res_{k'/k} A'$ is $k$-isogenous
to $A^n$, so this gives us an $R$-algebra 
isomorphism of $R\tensor_{R'}
  \End_{k'}(A')$ to a matrix algebra,
  $R\tensor_{R'}
  \End_{k'}(A')\stackrel{\sim}{\to} 
  M_n(\End_k^0(A))$ where $R$ is identified with the usual $\Q[\pi]
  \subset \End_k^0 A$ compatibly. The theorem then gives us
  \[F\tensor_{{F'}} \End_{k'}^0(\tilde{A'}) \sim
  \End_k^0(A)\,\,\text{  in }\Br(F)\] 
when $A$ is $k$-simple if two
properties hold: (i) $f_A$ a polynomial in $T^{[k':k]}$,
and (ii) there is a {\em unique} $k'$-simple factor  $\tilde{A'}$
of $k' \tensor_k A$.  Here, $F$ and $F'$ are the
$\Q$-subalgebras generated by the Frobenius automorphisms
  $\pi$ and $\pi'$ relative to $k$ and $k'$ on $A$
and $\tilde{A'}$ respectively.  In fact, (i) implies (ii)
for $k$-simple $A$.
Indeed, if $f_A = h(T^{[k':k]})$  then
$f_{k' \otimes_k A} = h^n$, and if moreover $A$ is $k$-simple
then $h = g^m$ for some $m \geq 1$ and irreducible $g \in \Q[T]$,
so $f_{k' \otimes_k A} = g^{mn}$.  This implies (ii), by
Theorem \ref{motivation}. 

\medskip

Now we turn to the second topic in this appendix, the
computation of the local invariant of certain cyclic algebras
over non-archimedean local fields.  It is this formula
that ultimately provides the explicit determination 
of the local invariants for the endomorphism
algebras of abelian varieties over finite fields.  
We first record a general lemma:

\begin{lemma}\label{reddiv} Let
$K$ be a field
and let $K'/K$ be a cyclic extension.
Fix a generator
$s'$ of $\Gamma = {\rm{Gal}}(K'/K)$.
Let $\chi_{s'}:\Gamma \rightarrow \Q/\Z$ be the unique
homomorphism sending $s'$ to $1/[K':K]$,
and let $\theta_{s'} = \delta(\chi_{s'}) \in
{\rm{H}}^2(\Gamma,\Z)$. 

\begin{enumerate}
\item For $c \in K^{\times}$, the class of the cyclic algebra 
$(K'/K,s',c)$ in ${\rm{Br}}(K)$
is the image of $c \bmod {\rm{N}}^{K'}_K({K'}^{\times})$
under the Tate periodicity isomorphism
$$K^{\times}/{\rm{N}}_{K}^{K'}({K'}^{\times}) =
\widehat{{\rm{H}}}^0(\Gamma,{K'}^{\times}) 
\stackrel{\cup \theta_s}{\simeq}
{\rm{H}}^2(\Gamma,{K'}^{\times}) \subseteq
{\rm{Br}}(K).$$
\item If $K'_0/K$ is a subextension 
and $s'_0 = s'|_{K'_0}$
then 
$$[(K'_0/K,s'_0,c)] = [(K'/K,s',c^{[K':K'_0]})]$$ in ${\rm{Br}}(K)$.
\end{enumerate}
\end{lemma}

\begin{proof}
The first part is an explicit cocycle calculation via
unwinding definitions and using the
cup-product compatibility of the $\delta$-functorial map
${\rm{H}}^{\bullet}(\Gamma,\cdot) \rightarrow
\widehat{{\rm{H}}}^{\bullet}(\Gamma,\cdot)$
(using Tate cohomology only in non-negative degrees).
The second part follows from
the first via the commutativity of the diagram
$$\xymatrix{
{K^{\times}/{\rm{N}}({K'_0}^{\times})} \ar[d] \ar[r]^-{\simeq} &
{{\rm{H}}^2(K'_0/K,{K'_0}^{\times})} \ar[d]^-{\rm{inf}}\\
{K^{\times}/{\rm{N}}({K'}^{\times})} \ar[r]_-{\simeq} &
{{\rm{H}}^2(K'/K,{K'}^{\times})}
}$$
(which in turn follows from the definitions
of the horizontal isomorphisms and both the $\delta$-functoriality 
and cup product compatibilities of inflation maps).
\end{proof}


Take $K$ to be a non-archimedean local field, and define
$${\rm{inv}}_K:{\rm{Br}}(K) \simeq \Q/\Z$$ as in
\S\ref{revcsa}.  By Lemma \ref{reddiv} and the existence
of unramified splitting fields for all Brauer
classes of $K$, it follows that every element of
${\rm{Br}}(K)$ is represented by a cyclic algebra
of the form $(K'/K,\phi,c)$ with $K'/K$ a finite
unramified extension, $\phi \in {\rm{Gal}}(K'/K)$
the {\em arithmetic Frobenius} generator, and $c \in K^{\times}$
an element.  (We could also work with geometric
Frobenius generators, but for the applications
to abelian varieties it is the cyclic algebras resting
on arithmetic Frobenius that naturally arise in the
analysis of the Dieudonn\'e modules.)
The main result is:

\begin{theorem}\label{localinv}
Let $K$ be a non-archimedean local field
and $K'/K$ an unramified finite extension.
Let $\phi \in {\rm{Gal}}(K'/K)$ be the arithmetic
Frobenius element. 
For any $c \in K^{\times}$,
the cyclic $K$-algebra $(K'/K,\phi,c)$ has
local invariant in $\Q/\Z$
represented by ${\rm{ord}}_K(c)/[K':K]$.
\end{theorem}

\begin{proof} 
The normalized valuation for $K$ identifies
$K^{\times}/{\rm{N}}^{K'}_K({K'}^{\times})$ with
$\Z/[K':K]\Z$,
so $[(K'/K,\phi,c)] \in {\rm{Br}}(K)$
only depends on the ratio ${\rm{ord}}_K(c)/[K':K]$. 
In particular, we may change $c$ by a unit
multiple so as to reduce to the case when $c$ is a power
of a local uniformizer of $K$, and by passing
to a suitable subextension over $K$ we may reduce
to the case when ${\rm{ord}}_K(c)$ is relatively prime to $[K':K]$. 
In this case the class of
$c$ in $K^{\times}/{\rm{N}}_K^{K'}({K'}^{\times})$
has order $[K':K]$ and so
the cyclic $K$-algebra $(K'/K,\phi,c)$ of
degree $[K':K]^2$ is necessarily a {\em division algebra}
(due to period dividing the index).   

A central division algebra $D$ over $K$
admits a unique valuation extending the
normalized valuation on $K$ \cite[Ch.~XII,~\S2]{Serre}, and there
is a classical procedure that uses this valuation to
compute ${\rm{inv}}_K([D])$ as follows.
Let $$\ord_D = \ord_K \circ {\rm{Nrd}}_D:D^{\times} 
\rightarrow \Z$$
(with ${\rm{Nrd}}_D$ denoting the reduced norm), so if
$[D:K] = n^2$ then 
$(1/n)\ord_D$ restricts to the normalized valuation on $K^{\times}$. 
Since we use the method of non-abelian cohomology 
(as in \cite[Ch.~X,~\S5]{Serre}) to identify
${\rm{H}}^2(L/K,L^{\times})$ with the subgroup of
classes in ${\rm{Br}}(K)$ split by a Galois extension
$L/K$ and we take arithmetic 
Frobenius as the preferred topological
generator for the Galois group of a finite field, 
after some
explicit unwinding of definitions and calculation
with 2-cocycles one finds
$${\rm{inv}}_K([D]) = -\frac{{\rm{ord}}_D(\gamma)}{n} = -
\frac{{\rm{ord}}_D(\gamma^n)}{n^2},$$
where (Skolem--Noether!) $\gamma \in D^{\times}$ satisfies 
$\gamma x \gamma^{-1} = \phi^{-1}(x)$
for $x$ in a copy of the degree-$n$ unramified extension
$K_n/K$ within $D$ (so $\gamma^n \in K_n^{\times}$).  Taking
$D = (K'/K,\phi, c)$ we thereby get $${\rm{inv}}_K([D]) = 
-\frac{{\rm{\ord}}_D(1/c)}{n^2} \bmod \Z = 
\frac{{\rm{ord}}_K(c)}{n} \bmod \Z.$$
\end{proof}


%(The application is for $F'=\Q$, $F=\Q(\sqrt{p})$.)
\end{rem}
%\bibliographystyle{alpha}
\begin{thebibliography}{Mum70}

\bibitem[AT68]{artintate}
E.~Artin and J.~Tate.
\newblock {\em Class field theory}.
\newblock Addison-Wesley, New York, 1968.

\bibitem[CF86]{Cassels}
J.~W.~S. Cassels and A.~Fr{\"o}hlich, editors.
\newblock {\em Algebraic number theory}, London, 1986. Academic Press Inc.
\newblock Reprint of the 1967 original.

\bibitem[Fon77]{Fontaine}
Jean-Marc Fontaine.
\newblock {\em Groupes {$p$}-divisibles sur les corps locaux}.
\newblock Soci\'et\'e Math\'ematique de France, Paris, 1977.
\newblock Ast\'erisque, No. 47-48.

\bibitem[Hon68]{Honda}
Taira Honda.
\newblock Isogeny classes of abelian varieties over finite fields.
\newblock {\em J. Math. Soc. Japan}, 20:83--95, 1968.

%\bibitem[Jac43]{Jacobson}
%Nathan Jacobson.
%\newblock {\em The {T}heory of {R}ings}.
%\newblock American Mathematical Society Mathematical Surveys, vol. I. American
%  Mathematical Society, New York, 1943.

\bibitem[Lam91]{Lam}
T.~Y. Lam.
\newblock {\em A first course in noncommutative rings}, volume 131 of {\em
  Graduate Texts in Mathematics}.
\newblock Springer-Verlag, New York, 1991.

%\bibitem[Man63]{Manin}
%Y. Manin.
%\newblock Theory of commutative formal groups over
%fields of finite characteristic.
%\newblock {\em Russian Math Surveys} 18:3--90, 1963.


\bibitem[MW71]{MiWa} J.~S. Milne and W.~C. Waterhouse.  \newblock
  Abelian varieties over finite fields.  \newblock In {\em 1969 Number
    Theory Institute (Proc. Sympos. Pure Math., Vol.  XX, State Univ.
    New York, Stony Brook, N.Y., 1969)}, pages 53--64. Amer.  Math.
  Soc., Providence, R.I., 1971.

  
\bibitem[Mum70]{Mumford} David Mumford.  \newblock {\em Abelian
    varieties}.  \newblock Tata Institute of Fundamental Research
  Studies in Mathematics, No. 5.  Published for the Tata Institute of
  Fundamental Research, Bombay, 1970.



\bibitem[Rei03]{Reiner}
I.~Reiner.
\newblock {\em Maximal orders}, volume~28 of {\em London Mathematical Society
  Monographs. New Series}.
\newblock The Clarendon Press Oxford University Press, Oxford, 2003.
\newblock Corrected reprint of the 1975 original, with a foreword by M. J.\
  Taylor.

\bibitem[Ser79]{Serre}
J-P.~Serre.
\newblock {\em Local fields}, volume 67 of
{\em  Graduate Texts in Mathematics}.
\newblock Springer-Verlag, New York, 1979.

\bibitem[Tat66]{TateInv}
John Tate.
\newblock Endomorphisms of abelian varieties over finite fields.
\newblock {\em Invent. Math.}, 2:134--144, 1966.

\bibitem[Tat68]{IsogenyClasses}
John Tate.
\newblock Classes d'isog\'enie des vari\'et\'es ab\'eliennes sur un corps fini.
\newblock {\em S\'eminaire Bourbaki}, 21:95--110, 1968.

\end{thebibliography}
%\bibliography{isogeny}
\end{document}














