Connection with the Birch and Swinnerton-Dyer Conjecture

Suppose $E$ is an elliptic curve over  $\mathbb{Q}$ and that $L(E,1)=0$. The Birch and Swinnerton-Dyer conjecture for $E$ asserts (among other things) that $E(\mathbb{Q})$ is infinite. Suppose $A$ is constructed as in Section 6. In this section we describe why if a certain consequence of a refinement of the Birch and Swinnerton-Dyer conjecture for $A$ is true, then $\Sel^{(n)}(E/\mathbb{Q})$ is nonzero.

Using modular symbols one sees that $L(A,1)\equiv 0 \pmod{\ell}$, so a refinement of the Birch and Swinnerton-Dyer formula for rank 0 abelian varieties predicts that there should be a nonzero element in $\ker ({\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\... ...ing{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A/E[n]))$. Thus by Proposition 8.2, either $H^1(X_{\text{\'et}},\mathcal{E})[n]\neq 0$, or there is a nonzero element of order dividing $n$ in

$$\ker(H^1(X_{\text{\'et}},\mathcal{A})\rightarrow H^1(X_{\text{\'et}},\mathcal{R})) \cong E(\mathbb{Q})/R(\mathbb{Q}),$$

in which case $E(\mathbb{Q})/R(\mathbb{Q})$ contains a nonzero element of order dividing $n$, so $E(\mathbb{Q})$ is infinite. Thus either ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E)[n]\neq 0$ or $E(\mathbb{Q})$ is infinite, so $\Sel^{(n)}(E/\mathbb{Q})$ is nonzero.