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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.
19
votes
Status of Fontaine-Mazur conjecture
It is true if $V$ is of dimension 1, essentially by class field theory (as you are considering only representations of $G_{\mathbb Q}$). Otherwise, it is still largely open for the following reasons.
…
- 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 …
- 10.9k
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 …
- 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
7
votes
Cohomology of $SL_2(\mathbb{F}_p)$ acting on trace zero matrices over $\mathbb{F}_p$
It seems to me that Sah's lemma will do the trick.
(Sah's lemma) Let $G$ be a group, $M$ a $G$-representation and $g\in Z(G)$. Then $x\mapsto (g-1)x$ is the zero map on $H^{1}(G,M)$.
The proof i …
- 10.9k
7
votes
3
answers
2k
views
Free subquotient of Galois representations coming from Hida theory
Let $\mathbf{T}$ be the reduced nearly ordinary Hecke algebra of level $N$ of Hida theory for $\operatorname{GL}_{2}$ over $\mathbb{Q}$ (or more generally over a totally real field $F$). Then $\mathbf …
- 10.9k
6
votes
Are Kato's zeta elements integral?
First, let us assume that $p$ is odd for safety.
Then I think your impression is correct: the first statement of 12.5 (4), i.e the integrality of the module generated by the $\mathbf{z}^{(p)}_{\gamma …
- 10.9k
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 …
- 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
5
votes
1
answer
590
views
What is the image of complex conjugation under Siegel Galois representations?
Let $G$ be the reductive group $\operatorname{GSp}_{4}$. Let $\pi$ be a smooth admissible cuspidal representation of $\operatorname{GSp}_{4}(\mathbb{A}^{(\infty)})$ of dominant weight. Assume, for cau …
- 10.9k
5
votes
Universal deformations of modular Galois representations
This is not always possible.
Suppose we could always find such an $M$. Take $\bar{\rho}$ reducible at $p$ with scalar image of the Frobenius. Then $M\otimes_{R_{\bar{\rho}}}R^{\operatorname{ord}}_{\ …
- 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
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 …
- 10.9k
4
votes
Accepted
CM abelian varieties and potential good reduction
No, absolutely not
In fact, the hypotheses you discuss are rather weak. Take $F$ a totally real number field. If $A/F$ is the abelian variety attached to an eigenform $f$ of weight $2$ and level $N$, …
- 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