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?

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    $\begingroup$ If $E$ has semistable (i.e. either good or bad multiplicative) reduction at $p$, then the corank conjecture is known: it follows from Kato's proof of one direction of the Iwasawa main conjecture, together with a non-vanishing result for $L$-values due to Rohrlich. The additive-reduction cases are nastier, and I don't know if the corank conjecture has been established in full generality. $\endgroup$ – David Loeffler Nov 18 '18 at 10:03

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.

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  • $\begingroup$ .. and it looks like it might even work for $p=2$ :) $\endgroup$ – Chris Wuthrich Nov 18 '18 at 14:26
  • $\begingroup$ Thanks David, the conjecture as stated is for a general number field, I wonder what the expected corank should be for the anticyclotomic and split prime $\mathbb{Z}_p$ extensions over an imaginary quadratic field $K$ in which $p$ splits into $\mathfrak{p}\mathfrak{p}^*$? The split prime $\mathbb{Z}_p$ extension at $\mathfrak{p}$ is the $\mathbb{Z}_p$ extension of $K$ ramified only at $\mathfrak{p}$. $\endgroup$ – user130124 Nov 18 '18 at 18:52
  • $\begingroup$ You should ask that as a new question. $\endgroup$ – David Loeffler Nov 18 '18 at 18:56
  • $\begingroup$ @DavidLoeffler I would like to understand Berger's argument (which I may have to adapt in a certain situation) where may I find this result? $\endgroup$ – user130124 May 11 '19 at 5:50
  • $\begingroup$ Laurent Berger, "Representations de de Rham et normes universelles", Bull. Soc. Math. France 133 (2005), no. 4, 601--618. $\endgroup$ – David Loeffler May 11 '19 at 9:05

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