Visualizing Mordell-Weil in Rank 0 Sha

Theorem 5.1   Let $E$ be an elliptic curve over  $\mathbb{Q}$. Conjecture 3.1 implies that for every rigid prime $p$, there is an abelian extension $K/\mathbb{Q}$ of degree $p$ such that

$$E(\mathbb{Q})/p E(\mathbb{Q}) \cong \Vis_J({\mbox{{\fontencoding{... ...ntfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A/\mathbb{Q})[p]),$$

where $J=\Res_{K/\mathbb{Q}} (E_K)$ and $A\subset J$ has dimension $p-1$ and rank 0.

Proof. Conjecture 3.1 produces a prime $\ell\equiv 1\pmod{p}$ such that $L(E,\chi_{p,\ell},1)\neq 0$ and $a_\ell(E)\not\equiv 2\pmod{p}$. Since $L(E,\chi_{p,\ell},1)\neq 0$ and $A$ is attached to $f\otimes \chi_{p,\ell}$, Kato's work implies that $A(\mathbb{Q})$ is finite. Lemma 4.1 implies that

$$p\nmid \char93 (J/E)(\mathbb{Q})_{\tor}\cdot \char93 E(\mathbb{Q}... ...Phi_{A,\ell}(\mathbb{F}_{\ell})\cdot \char93 \Phi_{E,\ell}(\mathbb{F}_{\ell}).$$

To apply Theorem 2.1, we just need that $p\nmid \char93 \Phi_{A,p}(\mathbb{F}_p)$. This is true becaue $\Phi_{A,p}(\overline{\mathbb{F}}_p) = \Phi_{A_K,\wp}(\overline{\mathbb{F}}_p)=0$, since $K/\mathbb{Q}$ is unramified at $p$, and $A_K=E_K\times \cdots \times E_K$ and $E$ has good reduction at $p$. Thus

$$E(\mathbb{Q})/p E(\mathbb{Q}) \cong \Vis_J({\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A)[p]),$$

as claimed.

$\qedsymbol$

BSD Connection: Let $E$ be an elliptic curve. Suppose we don't know anything about $E(\mathbb{Q})$, but do know that $L(E,1)=0$. If we could prove that there is a rigid prime such that

$${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontsh... ...hbb{Q})[p]\neq 0\qquad\text{(as {\em better be} predicted by the BSD formula)}$$

and

$${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/K)[p]=0,$$

then Theorem 5.1 would imply that $E(\mathbb{Q})$ is infinite.