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 $c \in \H ^1(K,A)$ is visible in some abelian variety $C_{/K}$ .

The next proposition gives evidence for elements of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, E)$ for an elliptic curve $E$ and elements of order 2 or 3.

Proposition 7.2.1   Suppose $E$ is an elliptic curve over $\mathbb{Q}$ . Then Conjecture 7.1.1 for $h = 0$ is true for all elements of order $2$ and $3$ in ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, E)$ .

Proof. We first show that there is an abelian variety $C$ of dimension $2$ and an injective homomorphism $i:E\hookrightarrow C$ such that $c\in {\mathrm{Vis}}_C({\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, E))$ . If $c$ has order $2$ , this follows from [AS02, Prop. 2.4] or [Kle01], and if $c$ has order $3$ , this follows from [Maz99, Cor. pg. 224]. The quotient $C/E$ is an elliptic curve, so $C$ is isogenous to a product of two elliptic curves. Thus by [BCDT01], $C$ is a quotient of $J_0(N)$ , for some $N$ . $\qedsymbol$

We also prove that Conjecture 7.1.1 is true with $h=1$ for all elements of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, A)$ which split over abelian extensions.

Proposition 7.2.2   Suppose $A_{/\mathbb{Q}}$ is a $J_1$ -modular abelian variety over  $\mathbb{Q}$ and $c \in {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q},A)$ splits over an abelian extension of  $\mathbb{Q}$ . Then Conjecture 7.1.1 is true for $c$ with $h=1$ .

Proof. Suppose $K$ is an abelian extension such that ${\mathrm{res}}_K(c)=0$ and let $C={\mathrm{Res}}_{K/\mathbb{Q}}(A_K)$ . Then $c$ is visible in $C$ (see Section 3.0.2). It remains to verify that $C$ is modular. As discussed in [Mil72, pg. 178], for any abelian variety $B$ over $K$ , we have an isomorphism of Tate modules

$${\mathrm{Tate}}_\ell({\mathrm{Res}}_{K/\mathbb{Q}}(B_K)) \cong {\mathrm{Ind}}^{G_\mathbb{Q}}_{G_K} {\mathrm{Tate}}_\ell(B_K),$$

and by Faltings's isogeny theorem [Fal86], the Tate module determines an abelian variety up to isogeny. Thus if $B=A_f$ is an abelian variety attached to a newform, then ${\mathrm{Res}}_{K/\mathbb{Q}}(B_K)$ is isogenous to a product of abelian varieties $A_{f^{\chi}}$ , where $\chi$ runs through Dirichlet characters attached to the abelian extension $K/\mathbb{Q}$ . Since $A$ is isogenous to a product of abelian varieties of the form $A_f$ (for various $f$ ), it follows that the restriction of scalars $C$ is modular. $\qedsymbol$

Remark 7.2.3   Suppose that $E$ is an elliptic curve and $c\in {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, E)$ . Is there an abelian extension $K/\mathbb{Q}$ such that ${\mathrm{res}}_K(c)=0$ ? The answer is ``yes'' if and only if there is a $K$ -rational point (with $K$ -abelian) on the locally trivial principal homogeneous space corresponding to $c$ (this homogenous space is a genus one curve). Recently, M. Ciperiani and A. Wiles proved that any genus one curve over $\mathbb{Q}$ which has local points everywhere and whose Jacobian is a semistable elliptic curve admits a point over a solvable extension of $\mathbb{Q}$ (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 $K/\mathbb{Q}$ is an abelian extension of prime degree then there is an exact sequence

$$0 \to A \to {\mathrm{Res}}_{K/\mathbb{Q}}(E_K) \xrightarrow{{\mathrm{Tr}}} E \to 0,$$

where $A$ is an abelian variety with $L(A_{/\mathbb{Q}},s) = \prod L(f_i,s)$ (here, the $f_i$ 's are the $G_\mathbb{Q}$ -conjugates of the twist of the newform $f_E$ attached to $E$ by the Dirichlet character associated to $K/\mathbb{Q}$ ). Thus one could approach the question in the previous remark by investigating whether or not $L(f_E,\chi,1)=0$ which one could do using modular symbols (see [CFK06]). The authors expect that $L$ -functions of twists of degree larger than three are very unlikely to vanish at $s=1$ (see [CFK06]), which suggests that in general, the question might have a negative answer for cohomology classes of order larger than $3$ .