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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 …

Suppose $K$ is an imaginary quadratic field (with class number 1, for simplicity), $p \ne 2$ a prime split in $K$, and $K_\infty$ the $\mathbb{Z}_p^2$-extension of $K$. If $E / \mathbb{Q}$ is an elli …

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 …

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 …

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 …

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 …

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 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 $ …

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 …

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.

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^ …

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 …

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 …