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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.
2
votes
Accepted
Do Tamagawa numbers of Galois representations stabilise in the cyclotomic tower?
As stated, the answer to your question is certainly no.
For instance, an elliptic curve $E/\mathbb Q$ with split multiplicative ordinary reduction at $p$ will have unbounded Tamagawa number at $p$ i …
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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 $N$, …
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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.
…
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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 …
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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 $p$- …
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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 …
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2
votes
Level lowering for weight 1 forms
Everything you wish for is true for modular forms over $\mathbb Q$ even at $p=2$, as it follows from refined forms of Serre's conjecture; here I am assuming of course that $\bar{\rho}$ is absolutely i …
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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 …
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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 …
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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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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 …
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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}}_{\ …
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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 …
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3
votes
Periods for 2-variable p-adic L-functions
Assuming you are writing about Mok's Compositio 2009 article, the answer is easy: it's a question of quantifier ordering. There are two statements which you could call the interpolation property for a …
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3
votes
Accepted
Why does $H^1(G_p/I_p,\mathbb{F}(\delta\epsilon^{-1}))$ vanish?
A variant of David Loeffler's answer...
If more generally $V$ is a $G_{\mathbb Q_{p}}$-representation with coefficients in a field, then the dimension of $H^{1}(G_{p}/I_{p},V)$ (EDIT : or rather $H^{ …
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