What is the current status on the corank conjecture for Selmer groups?

Question

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p$ a prime. It is conjectured in the book of Coates and Sujatha "Galois Cohomology of Elliptic Curves" (Conjecture 2.5) that the corank of the Selmer group of $E$ over the cyclotomic $\mathbb{Z}_p$ extension is $1$ when $E$ has potentially supersingular reduction at $p$ and $0$ otherwise. There is a really nice proof of this following Theorem 2.14 which is subsequently given in the case in which the Selmer group of $E$ over $\mathbb{Q}$ is finite (so in particular only when $E$ has rank zero and the $p$ part of the Tate Shafarevich group is assumed to be finite).

What is the current state of affairs on this conjecture?

Answer 1

Yes, the corank conjecture is a theorem for elliptic curves over $\mathbb{Q}$. The key to the proof is the following:

Theorem (Kato, 2004): For any $E$ and any $p$, the "fine Selmer group" $Sel_p^0(E / \mathbb{Q}_\infty) = \operatorname{ker}\Big(Sel_p(E / \mathbb{Q}_\infty) \to H^1(\mathbb{Q}_{p, \infty}, E[p^\infty])\Big)$ is cotorsion.

One of the great things about this theorem is that both its statement and its proof are completely independent of the local behaviour of $E$. The dependence on local behaviour comes when you try to use this to deduce things about the classical Selmer group $Sel_p(E / \mathbb{Q}_\infty)$ from Kato's theorem.

Combining Kato's theorem, Poitou--Tate duality, and a theorem in local Iwasawa theory due to Berger, one gets the following consequence:

Corollary: The corank of $Sel_p(E / \mathbb{Q}_\infty)$ is 1 if $T_p(E) |_{G_{\mathbb{Q}_p}}$ is irreducible, and 0 otherwise.

So one needs to check that $T_p(E) |_{G_{\mathbb{Q}_p}}$ is irreducible if and only if $E$ has potentially supersingular reduction, which is a fun exercise.