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An explicit formula is given at the bottom of page 21 of this PhD thesis.

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 reason why Hecke theory for $\Gamma(N)$ doesn't get much treatment in the literature is because you can easily reduce it to the $\Gamma_1(N)$ case. More precisely, you can conjugate $\Gamma(N)$ by …

I wouldn't recommend Beilinson's 1985 paper as a general reference -- it's terrifyingly compressed, developing an entire new subject in a single short paper, and crashes through the necessary material …

(Originally a comment, reposted as an answer:) Choose a primitive element $\alpha$ of F (i.e. such that $F=\mathbf{Q}(\alpha)$). Let $f$ be its minimal polynomial. Then the data of a field ordering o …

There is an excellent reason why the exponential term and the division by $n$ are there, although they look a bit mysterious at first. Firstly, a correction to your formula: it should be $|C(\mathbb …

I'm trying to find a reference for the following fact: If Tate's conjecture is true for all smooth projective varieties over $\mathbb{F}_p$, then the Frobenius endomorphism on the crystalline coho …

This is a very rich and active subject. There are lots of different approaches to the problem, giving more or less strong results -- you can try to interpolate any or all of { Hecke eigenvalues, Fouri …

This is much easier than it looks. The point is that any reductive group $G$ is isogenous to the product of its radical, which is its centre $Z(G)$, and its commutator subgroup, which is a semisimple …

It's easy to reduce to the case of computing the Petersson product of a normalised new eigenform with itself. Here you can use the fact that the product is equal to the value at s=k of the symmetric s …

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 …

There is a very nice introduction to Galois representations in chapter 9 of Diamond and Shurman's book "A First Course in Modular Forms". This is really thorough, e.g. it carefully explains the defini …

Let $p$ be a prime, and $F$ a $p$-adic Hida family of ordinary modular forms (of some tame level $N \ge 1$). I'd like to know whether, for $q$ a prime factor of $N$, the actions of the Atkin--Lehner i …

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 …

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 …