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?



What is the current status on the corank conjecture for Selmer groups?
Yes, the corank conjecture is a theorem for elliptic curves over $\mathbb{Q}$. The key to the proof is the following:



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> **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. 
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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:



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> **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.
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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.


