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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.
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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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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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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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 …
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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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