cokernel of $H^1(F_\Sigma/F,E[p^\infty])\to \prod_v H^1(F_v,E[p^\infty])/\operatorname{im}(\kappa_v)$

Question

Let $F$ be a number field and $E/F$ an elliptic curve. Fix an odd prime $p$.Let $\kappa:E(F)\otimes \mathbb Q_p/\mathbb Z_p\to H^1(F,E[p^\infty])$ the Kummer map and $\kappa_v$ its reduction.

Let $\gamma:H^1(F_\Sigma/F,E[p^\infty])\to \prod_v H^1(F_v,E[p^\infty])/\operatorname{im}(\kappa_v)$ where $\Sigma$ is the collection of the primes above $p,$ the bad primes and arch primes and the product runs over the primes of $\Sigma$

If $\operatorname{Sel}_{p^\infty}(E/K)$ is finite, i.e E is of rank 0 (assuming sha is finite), then Cassel's theorem says that $\operatorname{coker}(\gamma)\cong E(F)_p$. In particular, if $E(F)[p]=0$, the map is surjective.

Is anything similar true for positive rank? The only reference for this that I know is Greenberg's notes where he proves that the map will be surjective over $F_\infty$ (the cycl $\mathbb Z_p$ extension), for any rank (assuming the selmer is $\Lambda$-cotorsion, and without assuming anything on the torsion). Are there any conditions in which over number fields we can still say the map is surjective for any rank?

Answer 1

I will assume $p$ is odd to avoid having to think about places at infinity.

Cassels proved more: There is an exact sequence $$H^1\bigl(F_\Sigma/F,E[p^\infty]\bigr)\longrightarrow \bigoplus_{v\in \Sigma} H^1\bigl(F_v,E[p^\infty]\bigr)/im(\kappa_v) \longrightarrow S$$ where $S$ is the dual of the projective limit of the $p^n$-Selmer group. The map to $S$ is dual to the limit of the maps $$ \operatorname{Sel}_{p^k}(E/F) \longrightarrow \bigoplus_{v\in \Sigma} E(F_v)/p^k E(F_v) $$ (The local term here is dual to the one above by local Tate duality). The projective limit of the Selmer groups is a finitely generated $\mathbb{Z}_p$-module of rank equal to $\operatorname{rank}(E/F)+\operatorname{corank}(Ш(E/F))$ and the finite torsion subgroup is equal to the $p$-primary torsion of $E(F)$. If you assume that $Ш(E/F)$ is finite, the question becomes what is the kernel of the map from the $p$-adic completion $E(F)\otimes \mathbb{Z}_p$ to the sum of the $p$-adic completions of $E(F_v)$. For $v\nmid p$, these are finite groups, but not for $v\mid p$. A conjecture of Waldschmidt says that the image from the Selmer group to the local terms is as surjective as it is allowed to be, analogous to Leopoldt's conjecture for the $p$-adic regulator of units.

If you climb up a $\mathbb{Z}_p$-tower $F_{\infty}/F$, the chances that $\gamma$ becomes surjective increase because the cokernel will be dual to the the projective limit $\varprojlim_n \varprojlim_k \operatorname{Sel}_{p^k}(E/F_n)$ and, if for instance the rank stabilises in the tower or even better the usual Selmer group is $\Lambda$-torsion, then this projective limit tends to be zero.

All of this is a bit beyond Greenberg's introduction. The proof of Cassels theorem above can be deduced from the global duality results in "Cohomology of number fields".