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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^ …

(1) No, it depends on the maximal ideal (maximal ideals of $\Lambda$ biject with mod $p$ characters of $\Gamma$). (2) no longer makes sense. (3) It would suffice (for this particular step of Kato's pr …

Theorem (Xin Wan): If $E$ is an elliptic curve of square-free conductor $N$, and $p \ge 3$ is a prime such that $p \nmid N$ and $a_p(E) = 0$, then Kobayashi's $\pm$ Iwasawa main conjectures are tr …

Aha, an excuse to quote chunks of my most recent grant proposal :-) Iwasawa theory is heavily used in work on the BSD conjecture. For instance, the first positive result to be proved in the direction …

The distinction between the spaces of cusp forms and of all modular forms is not important for the Gouvea-Mazur conjecture, since it's very easy to show that the Eisenstein series vary in p-adic famil …

As far as I know, it is difficult to extract much information about the adjoint Selmer group over the cyclotomic $\mathbb{Z}_p$-extension. If the modular form $f$ corresponding to $\rho$ is ordinary a …

What's the "state of the art" in constructing multi-variable p-adic L-functions for number fields? More precisely: if K is a number field, and $K_{\infty} / K$ is an infinite Galois extension, unrami …

The formula you give relating $L_\alpha$ and $L_\beta$ is correct, but it is only valid for $1 \le j \le k-1$, so it only gives you finitely many values and hence it doesn't show that one L-function d …

(I was hoping somebody else would answer this, because function fields are not really my area and I hoped I would learn something from the answer; but nobody seems to be biting, so...) Iwasawa theory …

A flippant response is that people had the idea of using perfectoid theory in Iwasawa theory long before perfectoid theory even existed. What I'm referring to here is the work of Fontaine--Wintenberge …

This is a hard problem (and one which is easily overlooked by the unwary)! Just to be clear, I'll summarize (how I think about) the problem: as a relation between additive and multiplicative Fourier t …

Let $E$ be an elliptic curve and $\rho$ an Artin representation of $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$. Then there is a "twisted L-function" $L(E, \rho, s)$, corresponding to the …

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $T$ the $p$-adic Tate module of $E$. Kato's Euler system, constructed in the paper "P-adic Hodge theory and values of zeta functions of modular forms …

There is no reason why you shouldn't consider completed etale cohomology, instead of completed singular cohomology. If you look at Emerton's 2006 Inventiones paper which started the whole theory, he a …

Yes, that does indeed sound like something I might have said :) I was referring to some extremely powerful theorems, originally due to Michael Harris, which show that: The cohomology groups of automo …