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Some conjectures – like this one and also the BSD conjecture – are hard to find in precise form in a single place, because the community's understanding of statement of the conjecture changed over tim …

Are you asking for a proof of existence, or an explicit construction? These are very different things! It is immediate from the definition that there exists a finite family $(W_i, W_i')_{i \in I}$ wit …

Even for 0-dimensional varieties over number fields, the statement of Poitou–Tate duality is much more subtle than this: it's not enough just to compare the kernels of base-extension to $\overline{K}$ …

Any meromorphic modular function of weight $0$ for $\mathrm{SL}(2,\Bbb Z)$ is a rational function of $j$, say $P(j)$. Since your function is holomorphic, $P$ is a polynomial. Since your function has 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 …

Blasius and Rogawski's paper "Motives for Hilbert modular forms" (1993) proves a more general result for Hilbert modular forms over any totally-real field, which includes this as a special case.

The correct viewpoint is not "$\Lambda$ is like a disc", but "$\Lambda$ is like the functions on a disc". To see this, ask yourself: given an element $f \in \mathbb{Z}_p[[T]]$, what values can we plug …

I am sure I've already written an expository account of this somewhere, but I looked over the lecture and seminar notes on my webpage and couldn't find it, so I'll write one here instead. Suppose we s …

The condition shouldn't be "$n$ is prime" but "$n$ is either 1, 2, or a prime congruent to 3 mod 4". For instance $\mathbb{Q}(-5)$ has class number 2. The more general statement that the 2-torsion sub …

This is a special case of Pink's "canonical construction" functor, which associates various kinds of coefficient sheaves on a Shimura variety (etale $\ell$-adic sheaves, vector bundles with connection …

I found this old question while searching for Kato's paper myself. Just in case anyone else is also still looking for these, here's what I found. Kato's work was published in three installments in J. …

For a gentle(-ish) introduction to the "Ihara avoidance" method, you might want to consult the notes of Toby Gee's course on modularity lifting from the 2013 Arizona Winter School, www2.imperial.ac.uk …

The following theorem is due to Chris Skinner, in this 2014 paper. Let E/Q be an elliptic curve such that rank E(Q) = 1 and the Tate-Shafarevich group Sha(E / Q) is finite, and some other techni …

These operators certainly appeared in the 1970 paper by Atkin and Lehner: Atkin, A. O. L.; Lehner, J. Hecke operators on $\Gamma_0(m)$. Math. Ann. 185 (1970), 134–160. I don't know for sure th …

The statement you want follows fairly straightforwardly from Bass' conjecture -- sufficiently straightforwardly that it may well not have a separate name of its own. If $\Sigma$ is a sufficiently lar …