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Results tagged with galois-representations
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user 5015
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.
11
votes
Accepted
Are Kato's zeta elements integral?
There are two issues. Let $H=H^1_{\mathrm{Iw}}(\mathbb{Q},T)$ where $T=V_{\mathbb{Z}_p}(f)(1)$ and $f$ is the modular form associated to the isogeny class of $E$.
(1) What is $T$ ?
$T$ will correspo …
- 8,945
10
votes
Accepted
Best bounds toward Serre's uniformity conjecture
Since no specialist has replied to this question, I will add a long comment about the little I know.
The unconditional bound depending on the conductor which was used in the implementation in sage co …
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9
votes
In practice, how explicitly can we describe a Galois representation?
(Too long for a comment, but maybe not what you are looking for.) I think the Galois representations that you are interested in are those coming from geometry with coefficients in a $p$-adic field.
Fi …
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5
votes
Accepted
Semi-Simplicity of Mod-$\ell$ Galois Representations
Let us assume that $E$ has complex multiplication by a maximal order $\mathcal{O}$. Then $E[\ell]$ is a free rank $1$ module over $\mathcal{O}/\ell\mathcal{O}$. The image $G$ of $\rho_{E,\ell}$ is con …
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5
votes
Motivation for uniform surjectivity of mod l representations associated to elliptic curves
The conjecture is known for semistable curves. There the only non-surjective $\overline{\rho_p}$ come from isogenies and they have been treated by Mazur's theorem on $X_0(N)$.
When looking at the pro …
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5
votes
Accepted
Galois cohomology of Tate modules
Let $S$ be a finite set containing all places where an elliptic curve $E$ has bad reduction as well as $p$ and $\infty$. Write $T$ for $T_pE$ and $G_S$ for the Galois group of the maximal extension o …
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3
votes
Elliptic curve with CM and image of Galois representation in normalizer of nonsplit Cartan
By cm theory, $E[p]$ is isomorphic to $\mathcal{O}_K/(p)$ as an $\mathcal{O}_K$-module. Elements in $\operatorname{Gal}(\bar{\mathbb{Q}}/K)$ act $\mathcal{O}_K$-linearly on $E[p]$ since they commute w …
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