Tagged Questions

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.

246 views

Hodge-Tate weights of induced representation

Let $K$ be a finite extension of the field of $p$-adic numbers $\mathbb{Q}_p$ and let $E$ be another such extension, such that all the $\mathbb{Q}_p$ embeddings $K \to \bar{\mathbb{Q}}_p$ are ...
233 views

Representability of deformation functors via SGA

I'm trying to understand Böckle's proof of Theorem 2.1.1 in his notes on deformation theory. Let's start with some motivation. Let $\Gamma$ be a profinite group (I'm thinking of an absolute Galois ...
190 views

Some clarifications regarding Deligne's paper on $\ell$-adic representations arising from modular forms

I've posted this question few days ago on math.stackexchange because it seems quite superficial. However, since I've got no responses at all, I'm posting it here. If the question is not suitable, ...
107 views

Explicit deformations of pseudo representations

Let $G$ be a group (which I will be glad to consider to be the absolute Galois group of a $p$-adic field, and so satisfies Mazur's finiteness condition which appears in his paper Deforming Galois ...
215 views

extensions of crystalline representations

Denote by $G_p$ a choice of an absolute Galois group of $Q_p$, the field of $p$-adic numbers. Consider a continuous representations of $G_p$ on a $3$-dimensional $Q_p$ vector space that is a ...
222 views

integral p-adic Hodge theory and de Rham representations

$p$-adic Hodge theory gives us some comparison theorems between several cohomology theories. It also provides a hierarchy in the category of $p$-adic representations of the absolute Galois group of a ...
226 views

Robba ring and overconvergent (phi,Gamma)-modules

It is my understanding that that every $p$-adic representation of the absolute Galois group of a finite extension $K$ of $\mathbb{Q}_p$ can be described in term of its associated $(\varphi,\Gamma)$-...
234 views

152 views

490 views

Elliptic curve E and Galois representation

Assume that an elliptic curve $E$ over $\Bbb Q$ has a reducible mod $p$ representation. Then Q: Why is the semi-simplification of $E[p]$ the direct sum of ${\Bbb Z}/p{\Bbb Z}$ and $\mu_p$? Next Q:...
256 views

500 views

Is there a yoga of effectivity for motives and their realizations?

Inside the 'usual' categories of motives (Chow, Voevodsky ones) there are the categories of effective motives. Similarly, there are effective Hodge structures; they seem to be closely related with ...
165 views

Is there literature on approaching semisimplicity of l-adic cohomology using vanishing cycles

Let $K$ be a finitely generated field, and $X/K$ a smooth proper variety. Let $G$ denote the absolute Galois group $\mathrm{Gal}(\bar{K}/K)$. Let $\ell$ be a prime different from $\mathrm{char}(K)$. A ...
630 views

Modularity theorem for abelian varieties

There's alredy two posts on MO about the extension of modularity to elliptic curves over fields other than $\mathbb{Q}$ ([1], [2]), and another one about general algebraic varieties [3]. What is ...
216 views

Galois representation attached to $3$-torsion points of an elliptic curve

Let $E$ - Elliptic curve defined over ${\mathbb{Q}}$. $G_{\mathbb{Q}}$ - The absolute Galois group, $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ of $\mathbb{Q}$. $E[3]$ - $3$-torsion points ...
328 views

Example of a variety over a number field with non-semisimple Galois representation on $\ell$-adic cohomology

This question is inspired by the question: Example of non-projective variety with non-semisimple Frobenius action on etale cohomology? Let $K$ be a number field (or finitely generated field of ...
If $G$ is a unitary group in $n$ variables over $\mathbb Q$, attached to an hermitian form for an imaginary quadratic extension $E/\mathbb Q$ and if we suppose that the hermitian form is definite over ...