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The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields is an important tool in number theory.

4 votes

Motivation of the construction of $p$-adic period rings

How did we end up with the such complicated constructions of $B$? To add to Laurent's answer remark that "these rings did not, however, come out of nowhere", I believe that in the early 80s, Fontain …
Olivier's user avatar
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4 votes

Properties of Mod $\ell^m$ Galois representation associated to modular form

Write $L$ for the finite Galois extension of $\mathbb Q$ with Galois group $G_{\mathbb Q}/\operatorname{Ker}\rho_{F,v}^m$. Then $\rho_{F,v}^m(\operatorname{Frob}_p)$ is the identity in $\operatorname{ …
Olivier's user avatar
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8 votes
Accepted

Proving automorphy of the Galois representations of number fields without considering the re...

The canonical answer to that question is certainly the world of so called converse theorems, whose basic ideas go back to Hecke's remark that an holomorphic $L$-function satisfying a suitable function …
Alex B.'s user avatar
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1 vote

About the restriction of a modular representation to a decomposition subgroup

This hinges on what you mean exactly by explicit description. Here is what is happening. Let me write $N_f$ for the conductor of $f$. Fontaine defined a number of so-called period rings to study $p$- …
Olivier's user avatar
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5 votes
Accepted

Is there relationship between $\mu=0$ for an elliptic curve and the irreducibility of its re...

I think generalizing a conjecture we know sol little about is a risky business, but let me try to say something non-vacuous. First of all, I'm assuming that $E$ has good ordinary reduction (otherwis …
Olivier's user avatar
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4 votes

Conductor of Galois representation attached to newform

In fact much more than the equality of conductor is true: the local Galois representation $\rho_{F,\lambda}|G_{\mathbb Q_{p}}$ obtained by restricting $\rho_{F,\lambda}$ to the decomposition group at …
Olivier's user avatar
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2 votes
Accepted

Are there Galois representations associated with any regular algebraic cuspidal automorphic ...

In Motifs et formes automorphes: applications du principe de fonctorialité by Laurent Clozel (in Automorphic forms, Shimura varieties and $L$-functions Volume I (1990)), it is asked in 4.3.2 whether t …
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1 vote
Accepted

Steinberg components of local deformation rings

If I understand correctly, I believe that you want $r:\Gamma\longrightarrow\operatorname{GL}_2(R)$ to factor through $R^{\operatorname{St}}$ if $r$ is a non-trivial extension of $\beta$ by $\alpha$ wi …
Olivier's user avatar
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4 votes
Accepted

Local Galois representation associated to twist of modular form

I think it helps to put things in a larger perspective. To an eigencuspform $f$ and a prime number $\ell$ is attached on the one hand an irreducible, admissible representation $\pi(f)_{\ell}$ of $\op …
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4 votes
Accepted

Reference on a result on local Galois representation associated to classic modular form in p...

The three articles referenced presented in logical order of exposition are respectively Faltings, Gerd Hodge-Tate structures and modular forms Math. Ann. 278 (1987) Tsuji, Takeshi $p$-adic étale coh …
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12 votes

To what extent are modular parametrizations expected to generalize?

A natural generalization of the geometric modularity conjecture which is compatible with your formulation Do you expect some form of modularity to correspond to the existence of a map from some sp …
Olivier's user avatar
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1 vote

Does the $p$-part of the level of a newform appear in its attached $p$-adic representation?

The answer to the question in the title is yes, as explained in the last paragraph below. However, under a literal interpretation of "can" (implying actual feasibility), I believe the answer to the q …
Olivier's user avatar
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4 votes

Applications of Level Lowering

First, a small clarification: level-lowering tells you that a modular representation in level $Np$ occurs in level $N$ only if by occurs you mean "is congruent to modulo $p$". That said, my answer to …
Olivier's user avatar
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6 votes

Status of conjectures in Serre's 1969 expose on Galois representations on l-adic cohomology

Because I recently had to think about this, let me sum up the results I know about conjecture C5. This conjecture is known to hold for any $m\in\mathbb N$ if the dimension of $Y$ is less than 2 by Ta …
Olivier's user avatar
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12 votes

Iwasawa main conjectures vs Bloch-Kato conjectures

If I understand your question properly, then I think much is known. Let me sum up what I understand about this picture. First a short answer to your question. Contrary to what you ask for, it is not …
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