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

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?

• 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. – 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.

• .. and it looks like it might even work for $p=2$ :) – Chris Wuthrich Nov 18 '18 at 14:26
• 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 $\mathfrak{p}$. – user130124 Nov 18 '18 at 18:52
• You should ask that as a new question. – David Loeffler Nov 18 '18 at 18:56
• @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? – user130124 May 11 '19 at 5:50
• Laurent Berger, "Representations de de Rham et normes universelles", Bull. Soc. Math. France 133 (2005), no. 4, 601--618. – David Loeffler May 11 '19 at 9:05