Theoretical Evidence for the Conjectures
The first piece of theoretical evidence for Conjecture 7.1.1 is Remark 3.0.2, according
to which any cohomology class
is visible in some abelian variety
.
The next proposition gives evidence for elements of
for an elliptic curve
and elements
of order 2 or 3.
Proposition 7.2.1
Suppose
is an elliptic curve over
.
Then Conjecture 7.1.1 for
is true for all
elements of order
and
in
.
Proof.
We first show that there is an abelian
variety
![$ C$](img7.png)
of dimension
![$ 2$](img68.png)
and an injective homomorphism
![$ i:E\hookrightarrow C$](img562.png)
such that
![$ c\in {\mathrm{Vis}}_C({\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, E))$](img563.png)
.
If
![$ c$](img95.png)
has order
![$ 2$](img68.png)
, this follows from
[
AS02, Prop. 2.4] or
[
Kle01],
and if
![$ c$](img95.png)
has order
![$ 3$](img536.png)
, this follows from
[
Maz99, Cor. pg. 224].
The quotient
![$ C/E$](img564.png)
is an elliptic curve, so
![$ C$](img7.png)
is isogenous to a product
of two elliptic curves. Thus by
[
BCDT01],
![$ C$](img7.png)
is a quotient
of
![$ J_0(N)$](img20.png)
, for some
![$ N$](img13.png)
.
We also prove that Conjecture 7.1.1 is true with
for all elements of
which split over abelian extensions.
Proposition 7.2.2
Suppose
is a
-modular abelian variety over
and
splits over an abelian extension of
. Then
Conjecture 7.1.1 is true for
with
.
Proof.
Suppose
![$ K$](img6.png)
is an abelian extension such that
![$ {\mathrm{res}}_K(c)=0$](img567.png)
and let
![$ C={\mathrm{Res}}_{K/\mathbb{Q}}(A_K)$](img568.png)
.
Then
![$ c$](img95.png)
is visible in
![$ C$](img7.png)
(see Section
3.0.2). It remains to verify that
![$ C$](img7.png)
is modular. As discussed in [
Mil72, pg. 178], for any abelian variety
![$ B$](img113.png)
over
![$ K$](img6.png)
,
we have an isomorphism of Tate modules
and by Faltings's isogeny theorem [
Fal86], the Tate module determines an
abelian variety up to isogeny. Thus if
![$ B=A_f$](img570.png)
is an abelian variety attached to a
newform, then
![$ {\mathrm{Res}}_{K/\mathbb{Q}}(B_K)$](img571.png)
is isogenous to a product of
abelian varieties
![$ A_{f^{\chi}}$](img572.png)
, where
![$ \chi$](img573.png)
runs through
Dirichlet characters attached to the abelian extension
![$ K/\mathbb{Q}$](img574.png)
.
Since
![$ A$](img5.png)
is isogenous to a product of abelian varieties of the form
![$ A_f$](img54.png)
(for various
![$ f$](img12.png)
), it follows that the restriction of scalars
![$ C$](img7.png)
is modular.
Remark 7.2.3
Suppose that
![$ E$](img271.png)
is an elliptic curve and
![$ c\in {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, E)$](img575.png)
. Is there
an abelian extension
![$ K/\mathbb{Q}$](img574.png)
such that
![$ {\mathrm{res}}_K(c)=0$](img567.png)
? The answer is ``yes'' if and only if there is a
![$ K$](img6.png)
-rational point (with
![$ K$](img6.png)
-abelian) on the locally trivial principal homogeneous space corresponding to
![$ c$](img95.png)
(this homogenous space is a genus one curve). Recently, M. Ciperiani and A. Wiles proved that any genus one curve over
![$ \mathbb{Q}$](img17.png)
which
has local points everywhere and whose Jacobian is a semistable elliptic curve admits a point over a solvable
extension of
![$ \mathbb{Q}$](img17.png)
(see [
CW06]). Unfortunately, this paper does not answer our question about
the existence of abelian points.
Remark 7.2.4
As explained in [
Ste04], if
![$ K/\mathbb{Q}$](img574.png)
is an abelian extension of
prime degree then there is an exact sequence
where
![$ A$](img5.png)
is an abelian variety with
![$ L(A_{/\mathbb{Q}},s) = \prod L(f_i,s)$](img577.png)
(here, the
![$ f_i$](img60.png)
's
are the
![$ G_\mathbb{Q}$](img61.png)
-conjugates of the twist of the newform
![$ f_E$](img578.png)
attached to
![$ E$](img271.png)
by the Dirichlet character associated to
![$ K/\mathbb{Q}$](img574.png)
). Thus one could approach
the question in the previous remark by investigating whether or not
![$ L(f_E,\chi,1)=0$](img579.png)
which one could do using modular symbols (see [
CFK06]).
The authors expect that
![$ L$](img97.png)
-functions of twists of degree larger than three are very
unlikely to vanish at
![$ s=1$](img580.png)
(see [
CFK06]), which suggests that in general,
the question might have a negative answer for cohomology classes of order larger than
![$ 3$](img536.png)
.
William Stein
2006-06-21