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Results tagged with galois-representations
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user 2284
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 …
- 10.9k
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{ …
- 10.9k
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 …
- 10.9k
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 …
- 10.9k
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 …
- 10.9k
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 …
- 10.9k
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 …
- 10.9k
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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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 …
- 10.9k
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 …
- 10.9k
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 …
- 10.9k
2
votes
Accepted
Smoothness of Hecke algebras
It follows from deformation theory of Galois (pseudo-)representations that $\mathbb T(\Lambda)$ is a complete noetherian semilocal ring. The maximal ideals correspond to mod $p$ modular representat …
- 10.9k
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 …
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6
votes
1
answer
412
views
Computing an eigencuspform in $S_2(\Gamma_0(1776))$
Consider
$$\bar{\rho}:G_{\mathbb Q}\longrightarrow\operatorname{GL}_2(\mathbb F_7)$$
the residual 7-adic Galois representation attached to the elliptic curve $y^2=x^3+x^2-4x-4$ of conductor 48. Then …
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