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This tag is used if a reference is needed in a paper or textbook on a specific result.

24 votes

Introductory text on Galois representations

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 …
17 votes
Accepted

What is the Perrin-Riou logarithm (or regulator)?

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 …
David Loeffler's user avatar
17 votes
2 answers
2k views

Why does Tate's conjecture imply semisimplicity of crystalline Frobenius?

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 …
David Loeffler's user avatar
14 votes
Accepted

BSD conjecture for rank 1 elliptic curves

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 …
David Loeffler's user avatar
12 votes

Chow Groups of varieties over number fields

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 …
David Loeffler's user avatar
11 votes
1 answer
773 views

Atkin--Lehner operators in Hida theory

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 …
David Loeffler's user avatar
10 votes
Accepted

Blueprint of L-functions and need for introducing them ( Hasse-Weil L-functions )

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 …
David Loeffler's user avatar
10 votes
Accepted

Reference request for Hecke operators for principal congruence subgroup of modular group

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 …
David Loeffler's user avatar
9 votes

Reference request for Kato's paper: A generalization of local class field theory by using K ...

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. …
David Loeffler's user avatar
8 votes
Accepted

Geometric interpretation of Iwasawa algebras: $\mathbb{Z}_p[[T]]$ as a disk?

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 …
David Loeffler's user avatar
7 votes
Accepted

Good references for K-theory of modular curves?

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 …
David Loeffler's user avatar
7 votes
Accepted

Origin of definitions of ramified Hecke operators

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 …
David Loeffler's user avatar
6 votes
Accepted

Rational Characters of a reductive group have the same rank as split component

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 …
David Loeffler's user avatar
6 votes

Class number of imaginary quadratic fields

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 …
David Loeffler's user avatar
5 votes

Numerical evaluation of the Petersson product of elliptic modular forms

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 …

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