\documentstyle{conm-p}  %this is the one we will use ultimately
\leftheadtext{Kenneth A. Ribet}
\rightheadtext{Fields of Definition of Abelian Varieties}

\newcount\refCount
\def\newref#1 {\advance\refCount by 1
\expandafter\edef\csname#1\endcsname{\the\refCount}}
  
   \newref BROWN %Ken Brown
   \newref ELK % Elkies
   \newref MER % Merkuriev
   \newref MATHANN %Math Annales paper
   \newref KOR % Ribet Korea paper
   \newref CG  % Cohomologie Galoisienne
   \newref SHBOOK %Shimura's book
   \newref CFHO %Shimura on class fields and Hecke operators


\def\startproof{\demo{Proof}}


\def\endproof{{\unskip\nobreak\hfil\penalty50\hskip1em
  \hbox{}\nobreak\hfil$\blacksquare$%
  \parfillskip=0pt \finalhyphendemerits=0
  \par}\enddemo}
%\let\endproof\enddemo
\def\leqno(#1){\eqno{\hbox to .01pt{}\rlap{\rm \hskip -\displaywidth(#1)}}}


\let\tempbigwedge\bigwedge
\def\bigwedge{\tempbigwedge\limits}
\let\tempcirc=\circ
\def\circ{\mathord{\raise0.25ex\hbox{$\scriptscriptstyle\tempcirc$}}}
\def\isom{\buildrel\sim\over\to}
\mathchardef\dualchar="7014
\mathchardef\strangeoplus="1208
\def\dual{\dualchar}
\def\hidedual{\llap{$\dualchar$}}


\def\ssigma{{}^\sigma\mkern-\thinmuskip}
\def\degF{[F\:\Q]}
\def\ttau{{}^\tau\mkern-\thinmuskip}
\def\ggg{{}^g\mkern-\thinmuskip}
\def\gammasign{\gamma_{\pm}}
\def\Q{{\bold Q}}
\def\Z{{\bold Z}}
\def\C{{\bold C}}
\def\Qbar{\overline{\Q}}
\def\gammabar{\overline{\gamma}}
\def\Gal{\mathop{\roman{Gal}}\nolimits}
\def\cor{\mathop{\roman{cor}}\nolimits}
\def\GalQ{\Gal(\Qbar/\Q)}
\def\seteq{\mathrel{:=}}
\def\Matrix#1{{\bold M}(2,#1)}
\def\GL#1{{\bold{GL}}(2,#1)}
\def\GLTWO{\bold{GL}_2}
\def\PGL#1{{\bold{PGL}}(2,#1)}
\def\Lie{\mathop{\roman{Lie}}\nolimits}
\def\ker{\mathop{\roman{ker}}\nolimits}
\def\Ind{\mathop{\roman{Ind}}\nolimits}
\def\Res{\mathop{\roman{Res}}\nolimits}
\def\Hom{\mathop{\roman{Hom}}\nolimits}
\def\End{\mathop{\roman{End}}\nolimits}
\def\Br#1{\mathop{\roman{Br}}\nolimits(#1)}
\def\Kbar{\overline{K}}
\def\calX{{\Cal X}}
\def\calA{{\Cal A}}
\def\mutwo{\{\pm1\}}
\def\Qstar{\Q^*}
\def\P{\Q_{>0}}
\def\H{{\roman H}}

\topmatter

 \title
Fields of definition of abelian varieties with real multiplication
 \endtitle

 \author
Kenneth A. Ribet
 \endauthor

 \affil
University of California, Berkeley
 \endaffil
\address{UC Mathematics Department, Berkeley, CA  94720 USA}
\endaddress

\email ribet\@math.berkeley.edu \endemail


\thanks 
1991 AMS Subject Classification: 11F11, 11G10 (primary);
14K10, 14K15.
This manuscript is in final form.
It was written primarily
during the program on $L$-functions
at the Isaac Newton
Institute for Mathematical Sciences.
It is a pleasure to thank the
INIMS for its hospitality.
The author wishes to thank 
N.~Elkies as well as
W.~Raskind
and J.-L.~Colliot-Th\'el\`ene for helpful
e-mail correspondence.
This work was partially supported by NSF Grants DMS 88-06815
and DMS 93-06898.
\endthanks

\endtopmatter
\document
\head 1. Introduction\endhead %\noindent
Let $K$ be a field, and let $\Kbar$ be a
separable closure of~$K$.
Let $C$ be an elliptic curve over~$\Kbar$.
For each~$g$ in the Galois group
$G\seteq\Gal(\Kbar/K)$,
let $\ggg C$ be the elliptic curve
obtained by conjugating $C$
by~$g$.
One says that $C$ is
an elliptic
$K$-curve
if all the
elliptic curves $\ggg C$
are $\Kbar$-isogenous to~$C$.

Recall that
a subfield $L$ of~$\Kbar$ is said to be a
$(2,\ldots,2)$-extension of~$K$
if $L$ is a compositum of a finite number 
of quadratic extensions of~$K$
in~$\Kbar$.  The extension $L/K$ is then Galois, and $\Gal(L/K)$
is an elementary abelian 2-group.
Recently, N.~Elkies proved:

%\noindent
\proclaim{(1.1) Theorem {\rm (Elkies, \cite\ELK)}}
Let $C$ be an elliptic
$K$-curve over~$\Kbar$ with no complex multiplication.
Then $C$ is $\Kbar$-isogenous to an elliptic curve defined over
a $(2,\ldots,2)$-extension of~$K$.\endproclaim

%\noindent
In this article, we
present an approach
to~(1.1) which seems different from that of~Elkies.
At the same time,
we generalize~(1.1) to include higher-dimensional
analogues of elliptic $K$-curves with no complex
multiplication.  These are
abelian varieties $A$ over~$\Kbar$
whose endomorphism
algebras are totally real fields of dimension
$\dim(A)$.

For lack of a better term, we borrow the phrase
``Hilbert-Blumenthal
abelian varieties" 
to refer to abelian varieties whose endomorphism
algebras are totally real fields of maximal dimension.
Our use of this expression is a bit unusual.
Indeed,
in standard parlance,
a Hilbert-Blumenthal abelian variety relative to a
totally real number field~$F$
is an abelian variety $A$ over~$\Kbar$ which is
furnished with an action of the ring of integers $\Cal O$
of~$F$.
One requires that the Lie algebra $\Lie(A/\Kbar)$ be
free of rank~one over~$\Cal O\otimes\Kbar$, which acts
on~$\Lie(A/\Kbar)$ by functoriality; in particular,
this requirement forces the dimension of~$A$ and the
degree of~$F$ to be equal.
In this article,
we insist that $\End(A)\otimes\Q$ be equal to
(and not bigger than) a totally real field of dimension~$\dim(A)$.
On the other hand,
we do not require the full ring of integers of this field
to act on~$A$.

Suppose that $A$ is a Hilbert-Blumenthal abelian variety
in our sense, and let $F$ be the totally real number
field $\End(A)\otimes \Q$.
We say that $A$ is a $K$-Hilbert-Blumenthal abelian
variety (or ``$K$-HBAV") if~$\ggg A$
is $F$-equivariantly isogenous to~$A$ for all $g\in G$.
The equivariance refers to the evident isomorphism
$\varphi\mapsto\ggg\varphi$ between the endomorphism algebras
of~$A$ and of~$\ggg A$:  we demand that there be 
for each $g\in\Gal(\Kbar/K)$ 
an
isogeny $\mu_g\:\ggg A\to A$
which satisfies $\varphi\circ\mu_g = \ggg\varphi\circ\mu_g$
for all $\varphi\in F$.

%\noindent
\proclaim{(1.2) Theorem}
Suppose that $A$ is a $K$-HBAV.
Then $A$ is $F$-equivariantly isogenous to
a Hilbert-Blumenthal abelian
variety over a finite $(2,\ldots,2)$
extension of~$K$.\endproclaim

One motivation for proving Theorem~1.2 is the study of
Jacobians of modular curves.
Indeed, let $f$ be
a weight-two newform on the group $\Gamma_1(N)$, and let $X_f$ be the
abelian variety associated to~$f$ by Shimura's construction
\cite{\SHBOOK{\rm, Th.~7.14}}.
Thus $X_f$ is a $\Q$-simple factor of the abelian variety~$J_1(N)$.
If $f$ is a cusp form with complex multiplication, then $X_f$
becomes isogenous to a power of a CM elliptic curve over~$\Qbar$.
In the opposite case, Propositions
2.1--2.2 below show that
the $\Qbar$-simple
factors of~$X_f$ are then either $\Q$-HBAVs or quaterionic
analogues of $\Q$-HBAVs.  (It should be possible to prove a
version of Theorem~1.2 in the quaternionic case as well.)
The absolute decomposition of~$X_f$ is controlled by
coincidences between the Galois conjugates 
of~$f$ and twists of~$f$
by Dirichlet characters (see \cite\MATHANN).
From the point of view of~\cite\MATHANN, one sees that this
absolute decomposition is  achieved over the abelian extension of~$\Q$
cut out by the set of Dirichlet characters which intervene.
This extension is a $(2,\ldots,2)$-extension of~$\Q$ if $f$ has
trivial Nebentypus character, but not in general.
Nevertheless, Theorem~1.2 tells us that the absolute ``building blocks"
of~$X_f$ are defined over a $(2,\ldots,2)$-extension of~$\Q$ in
all cases.


We turn now to a discussion of the proof of Theorem~1.2 and
some associated results.
First of all, let us indicate how
Theorem~1.2 follows immediately from a series of results in~\S3.
As we will see,
if $A$ is a $K$-HBAV, then $A$
defines a class $\gamma$
in the cohomology group $\H^2(G,F^*)$ made from locally
constant cocycles $G\times G\longrightarrow F^*$ and
the trivial action
of~$G$ on~$F^*$.
Proposition~3.1 shows that
the class~$\gamma$
represents the obstruction to finding a Hilbert-Blumenthal
abelian variety over $K$ which is isogenous over $\Kbar$ to the
given one. Therefore, to prove~(1.2) is to show
that $\gamma$ becomes trivial
under the cohomological restriction map corresponding
to a base extension $K \rightsquigarrow K'$, where $K'$
is a $(2,\ldots,2)$-extension of~$K$.

Now, by Proposition~3.2,
$\gamma$ lies in the subgroup
$\H^2(G,F^*)[2]$ of~$\H^2(G,F^*)$ consisting of classes
of order at most~two.
On the other hand, Theorem~3.3 asserts that each element
of~$\H^2(G,F^*)[2]$ becomes 
trivial
under the  restriction map corresponding
to a base extension $K \rightsquigarrow K'$
of the desired type.  This completes our discussion of~(1.2).

Secondly, we wish to highlight
a technical
point that arises in applying Theorem~3.3 to~$\gamma$.
Namely,
let $P$ be the quotient $F^*/\mutwo$
so that we have an exact sequence of abelian groups
$$0 \to\mutwo\to F^*\to P\to 0.$$
This sequence is {\it split},
since the abelian group
$P$ is free
(Lemma~3.5).   Consequently,
$\H^2(G,F^*)[2]$ is a split extension
of~$\H^2(G,P)[2]$ by~$\H^2(G,\mutwo)$.
As will be seen in~\S3, there is an
elementary isomorphism $ \Hom(G,P/P^2)\isom\H^2(G,P)[2]$.
Hence we have
a split exact sequence
$$
 0\to \H^2(G,\mutwo) \to \H^2(G,F^*)[2] \to \Hom(G,P/P^2)\to 0.
$$
Call $\gammabar$ the image of~$\gamma$ in~$\Hom(G,P/P^2)$.
Then $\gammabar$ cuts out a $(2,\ldots,2)$-extension
of~$K$. This is
the extension $K_P$ of~$K$
in~$\Kbar$
such that $\Gal(\Kbar/K_P)$
is the kernel of~$\gammabar$.
Certainly, any extension of~$K$
in~$\Kbar$ which trivializes~$\gamma$ must contain~$K_P$.

In the case of elliptic curves
(i.e., the case $F=\Q$), Elkies
shows that $\gamma$ is trivialized by
the field~$K_P$.
This point, although admittedly technical, seems very
striking to us. The present article may be viewed as
an attempt to find a generalization of this phenomenon
to the Hilbert-Blumenthal case.

Such a
generalization
is presented in~\S4, where we introduce the
presumably superfluous requirement that
$K$ has characteristic~zero. (This hypothesis
does not intervene in~\cite\ELK.)
We show (in Corollary~4.5 below)
that
$\gamma$ is trivialized by~$K_P$ whenever
$\degF$ is~odd, and more generally
whenever there is an embedding
$F\hookrightarrow\Kbar$ for which
the degree
$[FK\:K]$ is~odd.
It would be interesting to determine whether this
hypothesis is necessary.

In the case where $\degF$ is odd, $F^*$ is canonically
a product $P\times\mutwo$.
Indeed,
$P$ may be identified with
the {\it subgroup\/} of~$F^*$ consisting of elements
with positive norm to~$\Q^*$.
Hence we have canonically
$$\H^2(G,F^*)[2] = \H^2(G,\mutwo)\times \Hom(G,P/P^2).$$
This suggests a study of the image $\gammasign$ of~$\gamma$
in the first factor $\H^2(G,\mutwo)$.
One may be
tempted to think that $\gammasign$ is trivial, which  would
certainly explain the vanishing of~$\gamma$ over~$K_P$.
Although $\gammasign$ is trivial for
the $\Q$-elliptic curves constructed
by~Shimura in~\cite\CFHO, there seem to be examples
where $\gammasign$ can be non-trivial.
One such example was communicated to the author by
E.~Pyle of Berkeley, California; here, $K=\Q$ and $A$ is a
$\Q$-elliptic curve.

\head 2. $K$-Hilbert-Blumenthal abelian varieties
\endhead %\noindent
Let $K$, $\Kbar$ and~$G$ be as above, and
let $F$ be a totally real number field.

We consider pairs
$(A,\iota)$, where $A$ is
an abelian variety of dimension~$\degF$ over~$\Kbar$, 
and where $\iota$
is an isomorphism $F \isom\Q\otimes\End(A)$;
such a pair will be called a Hilbert-Blumenthal
abelian variety.
% (or HBAV) over~$k$.
As mentioned above,
it would be more standard
to allow $\iota$ to be an {\it injection\/}
$F\hookrightarrow \Q\otimes\End(A)$; however,
the more restrictive
definition seems to be convenient
in what follows.
It should be stressed that our Hilbert-Blumenthal
abelian varieties are,
in particular, non-CM abelian varieties.

Abusing notation, we will generally write
$A$ for the pair $(A,\iota)$.
Moreover, we will frequently
view $A$ as an object in the category of abelian
varieties {\it up to isogeny\/} over~$\Kbar$.
In this category, isogenies of abelian varieties become
isomorphisms, and the endomorphism algebra usually denoted
$\Q\otimes\End(A)$ can be written more simply as~$\End(A)$.

Suppose that $g$ is an element of~$G$.  
Then $\ggg A$ admits a natural multiplication~$\ggg\iota$
by~$F$, so that $\ggg A$ is again a Hilbert-Blumenthal
abelian variety.
As mentioned in~\S1,
we say that $A$ is a $K$-HBAV if
there is
an $F$-equivariant
isomorphism $\mu_g\:\ggg A\isom A$ of
abelian varieties up to isogeny for each $g\in G$. 
Notice that a $K$-HBAV of dimension~one is
an elliptic $K$-curve with no complex multiplication.

To motivate the study of $K$-HBAVs, we record
some facts concerning $\Qbar$-simple
factors of abelian varieties over~$\Q$ with many
endomorphisms.  We begin with some terminology:
an abelian variety $A$ over~$\Qbar$ is said to be
a ``fake" Hilbert-Blumenthal abelian variety
if its endomorphism algebra is a quaternion division
algebra over a totally real field~$F$ and if
$\dim(A)=2\cdot\degF$.
Also, an abelian variety $C$ over~$\Q$ is said to
be of~$\GLTWO$-type if
$\Q\otimes\End_\Q(C)$
is a number field
of degree~$\dim(C)$. 
One knows that the
$\Q$-simple factors of the Jacobian
of a modular curve $X_1(N)$ are of $\GLTWO$-type.
Moreover, it is reasonable to conjecture that {\it all\/}
abelian varieties of~$\GLTWO$-type over~$\Q$
are $\Q$-simple factors of the Jacobian of some~$X_1(N)$,
cf.~\cite{\KOR{\rm, 4.4}}.  (Such a conjecture may
be viewed as a higher-dimensional analogue of the
conjecture of Taniyama and Shimura to the effect
that all elliptic curves over~$\Q$ are modular.)


%\noindent
\proclaim{(2.1) Proposition}
Suppose that $C$ is an abelian variety over~$\Q$
of $\GLTWO$-type.
Then
$C_{/\Qbar}$ is ``isotypical":
it is isogenous to a product
$A\times\cdots\times A$, where $A$ is a simple abelian
variety over~$\Qbar$.
Further, $A$ must be one of the
following: (i) an elliptic curve
with complex multiplication; (ii) a Hilbert-Blumenthal
abelian variety (for some totally real field $F$);
(iii) a ``fake" Hilbert-Blumenthal abelian variety.
\endproclaim

\startproof
Proposition~2.1 is implicit in
the discussion of~\S5 of~\cite\KOR\
(which is based principally on results of~G.~Shimura).
For completeness, we shall deduce the proposition from
some results included in~\cite\KOR.

First of all, if $C_{/\Qbar}$ contains any non-zero
abelian subvariety with complex multiplication, then
a result of Shimura 
(Proposition~1.5 of~\cite\CFHO)
implies that $C_{/\Qbar}$ is a power of an elliptic
curve with complex multiplication; thus, we are
in case~(i).
Assume instead that $C_{/\Qbar}$ has no non-zero abelian
subvariety with complex multiplication.
The number field $E\seteq\Q\otimes\End_{\Q}(C)$ is
then
its own commutant in~$\Cal X$, the algebra of all
endomorphisms of~$C$ over~$\Qbar$.
This implies that the center of~$\Cal X$ is contained
in~$E$: it is therefore a subfield $F$ of~$E$.
One shows that $F$ is in fact a
{\it totally real\/} number field \cite{\KOR{\rm, 5.4}}.
Since the center of~$\Cal X$ is a single field
(as opposed to a product of several fields),
$C$ is isotypical, as we claimed.
In fact, if we write $\Cal X$ as
${\roman M}(n,D)$, where $D$ is
a division algebra with center~$F$,
then 
$C_{/\Qbar}$ is isogenous to the $n$th power of a
simple abelian variety $A$ over~$\Qbar$ whose endomorphism
algebra is~$D$.

Let $t$ be such that $t^2$ is
the rank of~$D$ over~$F$.  Then
a short calculation, based on the fact that $E$
is a maximal commutative semisimple subalgebra
of~${\roman M}(n,D)$, establishes the formula
$\dim(A) = t\cdot\degF$.  
An argument exploiting the action of~$D$
on~$\H^1(A({\bold C}),\Q)$ shows that $t$ is at
most~2.
(See
the proof of Proposition~5.2 of~\cite\KOR\ for these
facts.)
If $t=1$, we are
in case~(ii), while if $t=2$ we are in case~(iii).
\endproof


\proclaim{(2.2) Proposition}
Suppose that $C$ is an abelian variety over~$\Q$ of
$\GLTWO$-type.
Let $A$ be a simple $\Q$-quotient of~$C$
whose endomorphism algebra is a totally real
field.  Then $A$ is a $\Q$-HBAV.
\endproclaim

\startproof
\kern0pt From what we have seen,
the abelian variety $C_{/\Qbar}$ is isogenous to
a product $A\times\cdots\times A$
(with, say, $n$ factors), and we are in
case~(ii).  
Using the analysis of~\cite\KOR, \S5 again,
we see that the center of the endomorphism
algebra
${\Cal X}$ of~$C_{/\Qbar}$ is a subalgebra of the
algebra of $\Q$-endomorphisms of~$C$.
If this center is~$F$, then $\Cal X$ is isomorphic
to~${\roman M}(n,F)$, and the endomorphism
algebra of~$A$ is~$F$.  After fixing an isomorphism
${\Cal X}\approx {\roman M}(n,F)$,
we may view $A$ as the
image of the matrix whose upper left-hand
corner entry is~1 and whose other entries are~0.
In this model, there is an obvious $F$-equivariant
isogeny
$\lambda\:C_{\Qbar} \to A^n$, given by the $n$ different
matrices with a single~1 in the first column and 0's elsewhere.

Let $g$ be an element of~$G$ and take $d\in F$.  Then
we have a commutative
diagram
$$ \matrix A^n\hfill &\buildrel \lambda\over\leftarrow &C\hfill
&=&\ggg C\hfill& \buildrel\ggg\lambda\over\to &\ggg A^n\hfill
\cr
           \downarrow  d &&\downarrow  d
	&&\downarrow  \ggg d &&\downarrow  \ggg d\cr
          A^n\hfill &\buildrel \lambda\over\leftarrow &C\hfill
	&=&\ggg C\hfill& \buildrel\ggg\lambda\over\to &\ggg A^n\hfill\cr
   \endmatrix$$
in which the central square
expresses the fact that $d$ is defined over~$\Q$.
Contracting it, we get a diagram
$$\matrix A^n\hfill& \isom& \ggg A^n \hfill\cr
  \downarrow  d &&\downarrow\ggg d\cr
   A^n\hfill& \isom& \ggg A^n \hfill\cr\endmatrix $$
in which the isomorphism $A^n\isom \ggg A^n$
is~$\ggg\lambda\circ\lambda^{-1}$.

For each pair of integers
$(i,j)$ with $1\le i,j \le n$,
we get a map $A\to \ggg A$ by composing
the following maps:
the inclusion $A\hookrightarrow A^n$ which uses the $i$th coordinate,
the isomorphism $A^n\approx \ggg A^n$, and the projection 
$\ggg A^n  \to \ggg A$ which uses the $j$th coordinate.
For some pair $(i,j)$, this map is non-zero, hence an isogeny.
(Recall that $A$ is simple.)  Let $\kappa$ be this isogeny.
Then we have
$$  \matrix A \hfill& \buildrel\kappa\over\to& \ggg A  \hfill\cr
  \downarrow  d &&\downarrow\ggg d\cr
   A \hfill& \buildrel\kappa\over\to& \ggg A  \hfill\cr\endmatrix
$$
as desired.
\endproof
\null
\head 3. The class $\gamma$
\endhead %\noindent
Let $A$ be a $K$-HBAV.
For each $g$, let $\mu_g$ be an $F$-equivariant isomorphism
up to isogeny
$\ggg A\to A$.  
We can, and do, assume that the collection $(\mu_g)$ has
been constructed from a
model $A_o$ of~$A$ over a finite extension~$L$ of~$K$
in such a way
that $\mu_g$ and $\mu_{g'}$ are the same map
$\ggg A\to A$ whenever 
$g$ and~$g'$ coincide on~$L$.
Thus the association $g\mapsto\mu_g$ is in an
obvious sense locally constant.
For each pair $\sigma,\tau\in G$, we note that 
$\mu_\sigma\circ\ssigma\mu_\tau\circ \mu_{\sigma\tau}^{-1}$
is an automorphism of~$A$ up to isogeny, and consequently
of the form $\iota(c(\sigma,\tau))$ with $c(\sigma,\tau)\in F^*$.
The map $(\sigma,\tau)\mapsto c(\sigma,\tau)$ is a continuous
two-cocycle on~$G$ with values in~$F^*$,
with $F^*$ being regarded as a trivial $G$-module.  The image $\gamma$
of~$c$
in~$\H^2(G,F^*)$ is independent of the choices of the~$\mu_\sigma$.

The following proposition is a mild
generalization of~\cite{\KOR{\rm, Th.~8.2}}.

\proclaim{(3.1) Proposition}
Let $E$ be an extension of~$K$ in~$\Kbar$; let $H=\Gal(\Kbar/E)$
be the corresponding closed subgroup of~$G$.
Then
the Hilbert-Blumenthal
abelian variety
$A$ is $F$-equivariantly isogenous to a Hilbert-Blumenthal
abelian variety
over~$E$
if and only if the class $\gamma$ 
lies in the kernel of the restriction map $\H^2(G,F^*)\to\H^2(H,F^*)$.
\endproclaim
\startproof
It suffices to prove the proposition in the case $E=K$, in which
case the assertion to be proved is that $\gamma=1$ if and only if
$A$ is ($F$-equivariantly) isogenous to a
Hilbert-Blumenthal
abelian variety over~$K$.


Suppose first
that there is a Hilbert-Blumenthal
abelian variety
$B$ over~$K$, together with an
isomorphism $\lambda\:A\isom B_{/\Kbar}$ of 
Hilbert-Blumenthal
abelian varieties
up to
isogeny over~$\Kbar$.
For each $g\in G$, we obtain an isomorphism $\ggg \lambda\:\ggg A\isom B$.
After setting $\mu_g \seteq \lambda^{-1}\circ\ggg\lambda$,
we find that $c$ is identically~1, so that $\gamma$ is~trivial.

Conversely, suppose that $\gamma=1$.  
Let $L$ and~$A_o$ be as above; to simplify notation, we shall write
$A$ for~$A_o$.
After replacing $L$ by a finite extension of~$L$, we can,
and do, assume that the $\mu_\sigma$ are defined over~$L$.
Further, the hypothesis that $\gamma=1$ means that
there is a locally constant function $\alpha\:G\to F^*$
so that $c(\sigma,\tau)=\alpha(\sigma)\alpha(\tau)/\alpha(
\sigma\tau)$; we enlarge $L$, if necessary so that $\alpha$
is defined modulo $\Gal(\Kbar/L)$, and so that $L$ is a
Galois extension of~$K$.  We then replace $\mu_g$ by
$(1/\alpha(g))\cdot\mu_g$ for each $g\in G$.
The new $\mu$'s satisfy $\mu_{\sigma\tau}=\mu_\sigma\ssigma
\mu_\tau$, and $\mu_g$ depends only on the image of~$g$
in the finite group $\Delta=\Gal(L/K)$.
Finally, $\mu_g$ may be viewed as an 
$F$-equivariant
isomorphism up to isogeny $\ggg A\isom A$
which is defined over~$L$.

Let $R$ be the abelian variety $\Res_{L/K} A$, where
``Res" denotes Weil's ``Restriction of scalars" functor.
In other words, $R$ 
represents the functor $C\mapsto\Hom(C_{/L},A)$
from the category of abelian varieties up to isogeny over~$K$
to the category of $\Q$-vector spaces.
Thus $R$  is an abelian variety over~$K$ which is
furnished with a structural
homomorphism (up to isogeny)
$\lambda:R_{/L}\to A$.
Given $C$ over~$K$ and a homomorphism (of abelian varieties
up to isogeny) $\varphi\:C_{/L}\to A$, there is a unique
homomorphism $\theta\:C\to R$ over~$K$ such that $\lambda\circ\theta
=\varphi$.

One constructs $R$ by considering the product $\prod_{g\in\Gal(L/K)}
\ggg A$ and using obvious patching data to descend this product to~$K$.
In particular, $R$ has
dimension
$[L\:K]\cdot\dim(A)$.  From this point of view,
the map $\lambda\:R_{/L}\to A$
is the projection of~$\prod_{g\in\Gal(L/K)}
\ggg A$ onto its factor~$A$.
Alternatively, given $R$ with its structural map~$\lambda$,
and an element $g$ of~$\Gal(L/K)$, we can view $\ggg\lambda$
as a map $R_{/L}\to\ggg A$.  One shows that the
map $R_{/L}\to\prod_{g\in\Gal(L/K)}
\ggg A$ induced by the
family of~$\ggg\lambda$
is an isomorphism.

By functoriality, %it is clear that 
$F$ acts on~$R$ over~$K$.
Thus we have $F\subseteq\calX$, where $\calX=\End_K(R)$.
Further,
the universal property satisfied by~$R$ makes it easy to
compute $\End_K(R)$ as a $\Q$-vector space.
Indeed, we have 
$$\End_K(R)=\Hom_K(R,R) = \Hom_L(R_{/L},A)
= \strangeoplus_g \Hom_L(\ggg A,A) = \strangeoplus_g F\cdot \mu_g.$$
Let $[g]$ be the element of~$\End_K(R)$ which corresponds
to $\mu_g$ in the $g$th factor of the direct sum.
Then $[g]$ is the unique element in~$\End_K(R)$
such that $\lambda\circ[g]= \mu_g\circ \ggg\lambda$.
The unicity implies that each $[g]$ commutes with the
action of~$F$ on~$R$.
A
short computation, based on the unicity and the formula
$\mu_{\sigma\tau}=\mu_\sigma\ssigma\mu_\tau$,
shows that $[\sigma][\tau]=[\sigma\tau]$
for $\sigma,\tau \in\Delta$.
Also, the $[g]$ commute with~$F$.
Hence $\calX$ is in the end the group algebra $F[\Delta]$.

The field $F$ is a direct summand of the algebra~$\calX=F[\Delta]$
via the inclusion $a\in F \mapsto a\cdot [1]$ and the augmentation
map $[g]\mapsto 1$.
Let $B$ be the subvariety of~$R$ corresponding
to the direct summand $F$ of~$\calX$.
Then $B$ 
is an abelian variety over~$K$ with 
an induced action of~$F$.  We claim that $B$ is a
Hilbert-Blumenthal
abelian variety over~$K$,
i.e., that $\dim(B)=\dim(A)$,
and that in fact $A$ and~$B$
are isomorphic Hilbert-Blumenthal
abelian varieties up to isogeny over~$L$.
To see this, we remark that
$R_{/L}$ is isogenous to a product of copies of~$A$
(since the $\ggg A$ are each isogenous to~$A$), so that
$B_{/L}$ is certainly isogenous to a product of some
number of copies of~$A$.
To determine the dimension of~$B$, we remark
that
$$F=\End_K(B)=\Hom_K(B,R)= \Hom_L(B_{/L},A).$$
Since $F$ is the endomorphism algebra of~$A$,
$B_{/L}$ is isogenous to~$A$.
\endproof

%\noindent
Suppose now
that $A$ is a $K$-HBAV over~$\Kbar$.
Fix a polarization
$\theta_A\:A\to A\dual$ of~$A$ as
an abelian variety.  The associated Rosati involution
of the endomorphism algebra of~$A$ may be viewed as a positive
involution of the totally real field~$F$
and therefore is forced to be the identity.

Let $B$ again be a 
Hilbert-Blumenthal
abelian variety
over~$\Kbar$, and fix a
polarization $\theta_B$ of~$B$.  Then for each $F$-equivariant
map $\mu\:B\to A$, we define the ``degree" $\deg(\mu)$
by the formula
$$ \deg(\mu) = \mu \circ \theta_B^{-1}\circ\mu\dual\circ\theta_A,$$
so that $\deg(\mu)$ is an element of the endomorphism algebra 
of~$A$.  Identifying this algebra with~$F$, we consider
that $\deg(\mu)$ is an element of~$F$.
If $\Kbar$ is a subfield of~$\C$,
then $\theta_A$ and~$\theta_B$
identify $\bigwedge^2_F \H_1(A(\C),\Q)$
and $\bigwedge^2_F \H_1(B(\C),\Q)$
with~$F$.  The number $\deg(\mu)$ is the element of~$F$
describing the map $\bigwedge^2_F \H_1(B(\C),\Q) \to
\bigwedge^2_F \H_1(A(\C),\Q)$ induced by~$\mu$.

If $\mu\:A\to A$ is given by
an element $c$ of~$F$, then
$\deg(\mu)=c^2$. Also,
it is easy to check that $\deg$ is multiplicative in the following
sense.  Suppose that $(C,\theta_C)$ is a third polarized 
Hilbert-Blumenthal
abelian variety,
and that $\lambda\:C\to B$ is $F$-equivariant.
Then $\deg(\mu\circ\lambda)=\deg(\mu)\deg(\lambda)$,
provided of course that the ``degrees" are computed
with respect
to a fixed set of polarizations.


\proclaim{(3.2) Proposition}
Let $A$ be a $K$-HBAV over~$\Kbar$.
Then the order of the
associated cohomology class $\gamma\in
\H^2(G,F^*)$ is at most~two.\endproclaim

\startproof
Fix a polarization $\theta\:A\to A\dual$ of~$A$ and
for each $g\in G$ let $\ggg\theta$ denote the associated
polarization of~$\ggg A$.
Choose a family $(\mu_g)$ as above, and let $c$
be the $F^*$-valued two-cocycle on~$G$ defined by this family.
For each $g$, let
$$ d_g \seteq  \mu_g \circ\ggg\theta^{-1}\circ\mu_g\hidedual\circ\theta$$
be the degree of~$\mu_g$, calculated with respect to the
polarizations $\theta$ and~$\ggg\theta$.
This element of~$F$ is canonical in the following sense:
if we replace $\theta$ by another polarization $\theta'$ of~$A$,
then $d_g$ will remain unchanged provided that we replace
$\ggg\theta$ by~$\ggg\theta'$.

By construction,
$$c(\sigma,\tau)=
\mu_\sigma\circ\ssigma\mu_\tau\circ \mu_{\sigma\tau}^{-1}$$
for $\sigma,\tau\in G$.
On taking degrees, we find the formula
$$ c(\sigma,\tau)^2 = {d_\sigma d_\tau \over d_{\sigma\tau}} $$
which expresses the square of~$c$ as a coboundary.
\endproof


\proclaim{(3.3) Theorem}
Let $\gamma$ be an element of order dividing
two in~$\H^2(G,F^*)$.
Then there is an open normal subgroup $H$ of~$G$
such that $G/H$ is an elementary abelian two-group
and such that the image of~$\gamma$ in~$\H^2(H,F^*)$
is~trivial.  In other words, $\gamma$ becomes trivial
after $K$ is replaced by a finite $(2,\ldots,2)$-extension
of~$K$.
\endproclaim
\startproof
Let $P=F^*/\mutwo$, so that there is a tautological
exact sequence
$$0 \to\mutwo\to F^*\to P\to 0.\leqno(3.4)$$
In the case where $\degF$ is~odd, there is a natural
splitting of this exact sequence: we may view $P$ as
the {\it subgroup\/} of~$F^*$ consisting of elements
with positive norm to~$\Q^*$.
In the general case, (3.4) is still split,
but apparently in no natural way. This follows directly
from:

\proclaim{(3.5) Lemma}
The group
$P$ is a free abelian group, of countable rank.\endproclaim

%\noindent
To prove the lemma, we
consider the map $\phi\:a\mapsto (a)$ which takes
an element $a$ of~$P$ to the fractional ideal of~$F$
generated by a lift of~$a$ to~$F^*$.
The image of~$\phi$ is a subgroup (of finite index) of
the group of all fractional ideals of~$F$; this latter group
is the free abelian group on the set of non-Archimedean
places of~$F$.  Hence the image of~$\phi$ is a free abelian
group, so that $P$ is abstractly isomorphic to
the direct sum of the image of~$\phi$ and the kernel of~$\phi$.
On the other hand, $\ker(\phi)$
is the group $U/\mutwo$, where $U$ is the group of units of~$F$.
According to Dirichlet's theorem,
$\ker(\phi)$ is a free abelian group of rank~$n{-}1$,
 where
$n=\degF$.  It follows that $P$ is the direct sum of
two free abelian groups, one finitely generated and one
countably generated.  This proves the lemma.

Returning to the proof of~(3.3),
we fix a splitting of~(3.4).  We shall assume that this is
the indicated natural splitting if $\degF$ is~odd.
The splitting fixes
an isomorphism of abelian groups $F^*\approx\mutwo\times P$.
This isomorphism induces in turn a decomposition
$$ \H^2(G,F^*)[2] \approx \H^2(G,\mutwo) \times \H^2(G,P)[2],$$
where the notation ``$[2]$" indicates the kernel of multiplication
by~2.  The element $\gamma$ of~$\H^2(G,F^*)$ is then the product
of its two projections $\gammasign\in\H^2(G,\mutwo)$ and
$\gammabar\in\H^2(G,P)[2]$.  The factor $\gammabar$ is 
independent of the chosen splitting of~(3.4):
it is the image of~$\gamma$ under the map on
cohomology induced by the quotient map $F^*\to P$.  
On the other hand,
the
``sign" component $\gammasign$ of~$\gamma$ depends on the
splitting.
We thus consider that
$\gammasign$
is defined intrinsically only in the case where $\degF$
is~odd.

To prove (3.3), we must show that
both $\gammasign$ and~$\gammabar$ become trivial when
$K$ is replaced by a $(2,\ldots,2)$-extension of~$K$.

To treat $\gammabar$, we consider the exact sequence
$$ 1\to P \buildrel x\mapsto x^2\over
{\relbar\joinrel\longrightarrow} P \to P/P^2\to 1;$$
note that $P$ is torsion free.  The associated
long exact cohomology sequence then gives an isomorphism
$$ \Hom(G,P/P^2)\isom\H^2(G,P)[2].$$
Suppose that $\gammabar$ corresponds to the homomorphism
$\varphi\:G\to P/P^2$.  Then $\ker(\varphi)$
is a subgroup of~$G$ which corresponds to a
$(2,\ldots,2)$-extension $K_P$ of~$K$.
It is clear that $\gammabar$ becomes trivial over
this extension.

It remains to split~$\gammasign$.
It is known that $\H^2(G,\mutwo)=0$ if $K$ has
characteristic~2, cf.~\cite{\CG{\rm, p.~II-5}}.  Assume, then,
that the characteristic of~$K$
is different from~2.  The group
$\H^2(G,\mutwo)$ may then be identified with~$\Br K[2]$,
where $\Br K$ is the Brauer group of~$K$.
A theorem of S.\hskip 0.1667em A.~Merkur'ev \cite\MER\
states that $\Br K[2]$
is generated by the classes of quaternion algebras
over~$K$.  Since each quaternion algebra over~$K$
is split by a quadratic extension of~$K$, it follows that
$\gammasign$ is split by a $(2,\ldots,2)$-extension of~$K$,
as required.
\endproof





%\demo{Proof of (1.2)}
%This theorem follows directly from (3.2)
%and~(3.3).
%\endproof


\head 4. The odd-dimensional situation
\endhead %\noindent
In this section, we assume
that $K$ has characteristic~0.
(An assumption concerning the parity of~$\degF$
will be made later.)

We begin with some preliminary
comments concerning
the class $\gamma$ defined by a $K$-HBAV.
First, we note that
the
tensor product $F\otimes_\Q K$
decomposes as some direct sum of fields
$\strangeoplus_{\omega\in\Omega}
K_\omega$, where the $K_\omega$ are finite
extensions of~$K$.
The index set $\Omega$ is the set 
$\Hom(F,\Kbar)$ of field
embeddings $F\hookrightarrow\Kbar$, modulo the action
of~$G$ on~$\Hom(F,\Kbar)$.

Next, suppose that $L$ is a finite extension of~$K$,
and consider $M\seteq L\otimes_K \Kbar$
as a $G$-module, with $G$ acting trivially on the first
factor.  Choose an embedding $\sigma$ of~$L$ into~$\Kbar$,
and let $H=\Gal(\Kbar/\sigma(L))$.
Then $L\otimes_K \Kbar$ is the induced representation
$\Ind_{H}^G\Kbar$.
Indeed, $M$
may be written as the
group of functions $f\:\Sigma\to\Kbar$, where $\Sigma$
is the set of embeddings $L\hookrightarrow \Kbar$.
In this optic, $G$ acts via $(g\cdot f)(\tau)=g(f(g^{-1}\tau))$
for $g\in G$ and~$\tau\in\Sigma$.
It is clear that $M=\oplus_{\tau\in\Sigma}M_\tau$, where
$M_\tau$
consists of those
functions which vanish outside of~$\tau$.
Further, the subgroups $M_\tau$ of~$M$ are permuted
transitively by~$G$, since $G$ acts transitively on~$\Sigma$.
The formula $M=\Ind_{H}^G\Kbar$
then follows by a well known criterion (see, e.g.,
\cite{\BROWN{\rm, Ch.~III, 5.4}}).

Finally, we consider $F\otimes_\Q \Kbar$ as a $G$-module,
with $G$ acting trivially on the first factor and in the
natural way on the second.  Then
$$ F\otimes_\Q \Kbar = (F \otimes_\Q K) \otimes_K \Kbar
  = \strangeoplus_\omega (K_\omega \otimes_K \Kbar).$$
Therefore,
$$ F\otimes_\Q \Kbar = \strangeoplus_\omega \Ind_{H_\omega}^G \Kbar.$$
In this latter formula, we have chosen 
for each~$\omega$
a $K$-embedding
$K_\omega\hookrightarrow \Kbar$ and have put
$H_\omega\seteq \Gal(\Kbar/K_\omega)$.

In a similar vein, one proves
$$ (F\otimes_\Q \Kbar)^* = \strangeoplus_\omega \Ind_{H_\omega}^G \Kbar^*.$$
Shapiro's lemma \cite{\BROWN{\rm, Ch.~III, 6.2}} provides
an identification
$$ \H^i(G,\Ind_{H_\omega}^G \Kbar^*) = \H^i(H_\omega,\Kbar^*)$$
for each~$i\ge0$.  (The induced module
$\Ind_{H_\omega}^G \Kbar^*$ may
be regarded as a co-induced module because 
the index of~$H_\omega$ in~$G$ is finite.)
Thus
$$ \H^i(G,(F\otimes_\Q \Kbar)^*) = \strangeoplus_\omega
   \H^i(H_\omega,\Kbar^*)$$
for each~$i$.
Consequently,
$$ \H^1(G,F\otimes_\Q \Kbar)^*) = 0$$
by Hilbert's Theorem~90.
Similarly, 
$$  \H^2(G,(F\otimes_\Q \Kbar)^*) = \strangeoplus_\omega
 \Br{K_\omega} \leqno(4.1) $$
because $\H^2(H_\omega,\Kbar^*)$ is the Brauer group
of~$K_\omega$.

Now
consider the exact sequence of
$G$-modules
$$ 0\to F^* \to (F\otimes\Kbar)^* \to (F\otimes\Kbar)^*/F^* \to 0,
\leqno(4.2)$$
where $G$ acts in the usual way on~$\Kbar$ and
the indicated tensor products are taken over~$\Q$.
Exploiting the vanishing of~$\H^1(G,(F\otimes\Kbar)^*)$,
we obtain
$$ 0 \to \H^1(G,(F\otimes\Kbar)^*/F^*) \buildrel\delta\over\to
  \H^2(G,F^*) \to \H^2(G,(F\otimes\Kbar)^*) \to \cdots.$$
Here, $\delta$ is the indicated connecting homomorphism
in the long cohomology sequence arising from~(4.2).


\proclaim{(4.3) Lemma}
Let $A$ be a $K$-HBAV.
Then the element $\gamma$ of~$\H^2(G,F^*)$
defined by~$A$ lies in the image of~$\delta$.
Equivalently, the image of~$\gamma$
in~$\H^2(G,(F\otimes\Kbar)^*)$ is~zero.
\endproclaim
\startproof
We must exhibit an element $\beta$
of~$\H^1(G,(F\otimes\Kbar)^*/F^*)$ such that $\gamma=\delta(\beta)$.
Let $V=\Lie(A/\Kbar)$.  For each $g \in G$, the map $\mu_g\:\ggg A
\to A$ induces a $(F\otimes\Kbar)$-linear homomorphism
$\Lie(\ggg A/\Kbar)\to\Lie(A/\Kbar)$, or equivalently an $F$-linear
homomorphism $\lambda_g\:V \to V$ which is $g$-linear in the sense
that it satisfies $\lambda(a\cdot v) = g(a)\lambda(v)$ for
$a\in\Kbar$ and $v\in V$.
Now it is well known, and easy to verify, that the Lie algebra
$\Lie(A/\Kbar)$ is free of rank~one over~$F\otimes\Kbar$.
Let $v$ be
a basis 
of~$V$, considered as a
free rank-one $F\otimes \Kbar$-module.
Then one has $\lambda_g(v)=a_g\cdot v$ for some element
$a_g$ in~$(F\otimes\Kbar)^*$.  The relations among the $\mu_g$
provide the formula
$c(\sigma,\tau)a_{\sigma\tau}=a_\sigma\ssigma a_\tau$ for
$\sigma,\tau\in G$.
It follows that the function $G\to (F\otimes\Kbar)^*/F^*$ induced
by~$g\mapsto a_g$ is a 1-cocycle, and that the corresponding
class $\beta$ in~$\H^1(G,(F\otimes\Kbar)^*)$ maps to~$\gamma$
under~$\delta$.\endproof
% foom!

\proclaim{(4.4) Proposition}
Suppose that $\gamma$ 
is the element of~$\H^2(G,F^*)$ arising from a $K$-HBAV
and that $\gammabar=0$.
Then $\gamma=0$
provided that 
no element of order~two in the Brauer group of~$K$
is split by all extensions of~$K$ of the form $K_\omega$.
\endproclaim
%
\startproof
In view of the hypothesis $\gammabar=0$, we have
$\gamma=\gammasign\in\H^2(G,\mutwo)$.
According to ~(4.3), $\gamma$ lies in the kernel
of the map 
$$j\:\H^2(G,\mutwo) \to \H^2(G,(F\otimes\Kbar)^*)$$
induced by the inclusion of~$\mutwo$ in~$(F\otimes\Kbar)^*$.
This map may be viewed as the natural map
$$ \Br K[2] \to \strangeoplus_\omega \Br{K_\omega},$$
which is injective by hypothesis.
Thus $\gamma$ is indeed~0.\endproof

\proclaim{(4.5) Corollary}
Suppose that $\gamma\in H^2(G,F^*)$
arises from a $K$-HBAV.
Suppose that $\degF$ is~odd.
Then the class $\gamma$ becomes~zero
after $K$ is replaced by the extension $K_P$
of~$K$ defined by~$\gammabar$.
In particular,
if $\gammabar=0$, then $\gamma=0$.
\endproclaim

\startproof
The two assertions of the corollary
are equivalent, since $\gammabar$ becomes trivial
after $K$ is replaced by~$K_P$.
Because of~(4.4), to prove the second
assertion it is enough to prove that
the map $j$ which occurs in the proof of~(4.4) is injective
whenever $\degF$ is odd.

However, $\degF = \sum_\omega [K_\omega\:K]$.
Thus, if $\degF$ is~odd, then there is at least one
index~$\omega$ for which $[K_\omega\:K]$ is~odd.
Further, if $[K_\omega\:K]$ is odd, then it is
evident that the map $\Br K[2] \to \Br{K_\omega}$
is injective since there is a
corestriction
map $\cor\:\Br{K_\omega}\to\Br K$ whose composition with the
natural map $\Br K\to \Br{K_\omega}$ is multiplication
by~$[K_\omega\:K]$.
\endproof

\Refs
\catcode`\?=\active
\def?{.\hskip 0.1667em\relax}

\ref\no\BROWN\by
K?S. Brown\book Cohomology of groups
\bookinfo Graduate Texts in Math., vol. 87
\publ Springer-Verlag \publaddr New York, Heidelberg and Berlin
\yr1982\endref

\ref\no\ELK\by N. Elkies\paper
Remarks on elliptic $K$-curves\paperinfo
Preprint, 1993
\endref

\ref\no\MER\by A?S. Merkur'ev\paper
On the norm residue symbol of degree~2
\jour Dokl. Akad. Nauk. SSSR
\vol261
\yr1981
\pages542--547
\transl
Soviet Math. Dokl.\vol24\yr1981\pages546--551
\endref

\ref\no\MATHANN\by   K?A. Ribet
\paper Twists of modular forms and endomorphisms of
abelian varieties\jour Math. Ann.\vol 253\yr1980\pages 43--62\endref

\ref\no\KOR\bysame \paper
Abelian varieties over $\Q$ and modular forms
\inbook
1992 Proceedings
of KAIST Mathematics Workshop
\publaddr Taejon
\publ Korea Advanced Institute of Science and Technology
\yr 1992\pages53--79
\endref

\ref\no\CG\by J-P. Serre
\book Cohomologie galoisienne
\bookinfo Lecture Notes in Math., vol. 5, $4^{\roman e}$ \'edition
\publ
   Springer-Verlag\publaddr Berlin and New York
   \yr1973
   \endref

\ref\no\SHBOOK\by G. Shimura
\book
Introduction to the arithmetic theory of automorphic functions
\publ
   Princeton University Press
\publaddr Princeton \yr1971
\endref


\ref\no\CFHO\bysame
\paper Class fields over real quadratic fields and
Hecke operators
\jour Ann. of Math.\vol95\yr1972\pages131--190\endref

\endRefs
\enddocument

