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I think your question already contains its own answer. In classical, "vertical" Iwasawa theory one studies class groups, or other arithetic widgets like elliptic curve Selmer groups, in a limit over $ …

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 …

(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 …

Symplectic case: Here are two reasons (not necessarily the only ones) why $\operatorname{GSp}_{2n}$ is more convenient to work with than $\operatorname{Sp}_{2n}$. Firstly: there is no Shimura datum w …

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 …

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 …

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 …

Let me explain a bit more what that footnote was supposed to mean. As I'm sure you know, an Euler system for a Galois representation $V$ over a number field $K$ consists of a bunch of classes in $H^1 …

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 …

I do not think there is a reference for this theory, because as far as I know no such theory exists. I have spent a substantial portion of my career studying the arithmetic of the special values of th …

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 …

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^ …

(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 …

Yes, the main conjecture is isogeny-invariant. See here: B. Perrin-Riou, Variation de la fonction $L$ $p$-adique par isogénie, Algebraic number theory, Adv. Stud. Pure Math. 17 (1989), pp. 347-358.

The main conjecture is a theorem if the image of the mod $p$ Galois representation of E is the whole of $GL_2(\mathbf{F}_p)$. The full statement of the conjecture, which implies what you wrote about l …