The galois-representations tag has no wiki summary.

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### 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 ...

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### Hodge–Tate structures of modular forms

The title refers to the paper of Faltings:
Hodge-Tate structures and modular forms.
Math. Ann. 278 (1987), no. 1-4, 133–149.
The main theorem in the paper says that the associated Galois rep to a ...

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### Which degree does a motivic Galois representation show up in?

Consider a representation $\rho: \operatorname{Gal} (\overline{\mathbb Q} | \mathbb Q ) \to GL_n ( \overline{\mathbb Q}_\ell)$ that is a subrepresentation of $H^i(X, \overline{\mathbb Q}_\ell (j))$ ...

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### Lifting projective Galois representation with condition

Let $\bar{\rho}: G_K\to PGL_n(\mathbb{C})$ be projective representation of the absolute Galois group of a number field $K$ and $\varphi\in Aut(G_K)$.
A theorem of Tate tells us that we can always ...

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### Status of Fontaine-Mazur conjecture

In the language of Richard Taylor's 2004 (extended) ICM article (''Galois Representations'', Annales de la faculté des sciences de Toulouse (2004) Tome XIII, no. 1, 73-119), the conjecture is the ...

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### N-th root of unity in N-th division field of abelian variety?

Let K be a number field and A/K an abelian variety over it. Can it be that K(A[n]) does not contain a primitive n-th rooth of unity? If the answer is yes is it always possible to bound the bad n ...

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### Level-Lowering in Weight 2

Let $N$ and $p$ be relatively prime integers with $p$ a prime. Suppose $f$ is a weight $k=2$ (normalized, cuspidal, etc) newform of level $\Gamma_1(N) \cap \Gamma_0(p)$. I seem to recall the existence ...

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### References for 'Theory of $p$-adic Galois Representations by Fontaine & Ouyang'

Presently I am reading the 'Theory of $p$-adic Galois Representations by Fontaine & Ouyang'. I am finding it difficult for eg. the initial sections on $l$-adic geometric representation of finite ...

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### Elliptic curves and supercuspidal representations of conductor $p^2$

Let $E$ be an elliptic curve defined over $\mathbf{Q}$. Let $p \geq 5$ be a prime of additive reduction for $E$.
Let $f$ be the newform associated to $E$, and let $\pi$ be the irreducible admissible ...

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### n torsion groups of quadratic twists of elliptic curves

If $E$ is an elliptic curve over a number field $K$ and $E^{F}$ is a quadratic twist of $E$. Then it is stated in ``Ranks of twists of elliptic curves and Hilbert’s tenth problem" due to Mazur and ...

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### What are the strongest conjectured uniform versions of Serre's Open Image Theorem?

This question concerns the uniform conjectured effective versions and generalizations of these two results of Serre on $\ell$-adic Galois representations $\rho_{E,\ell}$ associated to a non-CM ...

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### 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 ...

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### 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 ...

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### 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, ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### Extensions of an abelian variety by a torus vs. extensions of their $\ell$-adic Tate modules

Let $K$ be a number field, let $A$ be an abelian variety over $K$, and let $H$ be a torus over $K$. For a prime $l$, we have the natural map
$$\mathrm{Ext}^1(A, H) \otimes_{\mathbb{Z}} \mathbb{Z}_l ...

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### Relation between cyclotomic character and fundamental character of level

The question I have is the following: is it true that the $p+1$ exponentiation of the fundamental character of level $2$ gives us the reduction (mod $p$) of the cyclotomic character?
For a review of ...

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### abelian $\ell$-adic representations in $\widehat{SL(2,Z)}$

In Grothendieck's Esquisse he claims that the action of
$$\text{Gal}(\mathbb Q)\to\text{Out}(\pi_1(M_{1,1})=\text{Out}(\widehat{SL(2,Z)})$$
obtained from the homotopy exact sequence of the étale ...

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### Are there known cases of the Mumford–Tate conjecture that do not use Abelian varieties?

(For a formulation of the Mumford–Tate conjecture, see below.)
The question
As far as I know, all non-trivial known cases of the Mumford–Tate conjecture more or less depend on the Mumford–Tate ...

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### Eichler-Shimura congruence

I'm trying to understand the Eichler-Shimura congruence which relates the Hecke operator $T_p$ to Frobenius at $p$ in characteristic $p$.
Two possible ways to compute $T_p$ mod $p$ seem to be:
A) ...

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### Any counterexamples known for the Generalized Tate conjecture?

One can state the generalized Tate conjecture over arbitrary finitely generated fields; to this end one should just define Galois representation to be effective if the eignevalues of the actions of ...

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### Tate's conjecture and symmetry of Hodge-Tate weights

I'm reading Bellaiche's notes on the Block-Kato conjecture (Hawaii summer school). Here is the link http://people.brandeis.edu/~jbellaic/BKHawaii5.pdf
At page 10 he claims that an indirect ...

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### Adelic open image for modular forms?

There's a famous theorem of Serre that if $E$ is a non-CM elliptic curve over $\mathbf{Q}$, and $\rho_{E, \ell} : Gal(\overline{\mathbf{Q}}/{\mathbf{Q}}) \to GL_2(\mathbf{Z}_\ell)$ is its $\ell$-adic ...

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### Rank one (phi,Gamma)-modules

Let $\mathbb{Q}_p$ be the field of $p$-adic numbers. Consider an unramified representation $\rho : Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p) \to \mathbb{F}_p^{\times}$ which sends the arithmetic ...

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### Field of definition of Galois representations of weight 1 modular forms

Let $f$ be a weight 1 modular form (let's say cuspidal, new, normalized, and a Hecke eigenform). Then there's an associated Artin representation $\rho_f: \operatorname{Gal}(\overline{\mathbf{Q}} / ...

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### Invariant obstructions to gluing Galois representations on elliptic curves

Let $E$ be an elliptic curve over $\overline{\mathbb F_p}$, or another separably closed field of characteristic $p$. Let $K$ be the function field of $E$, and let $K_p$ be the local field at a point.
...

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### Applications of anabelian geometry to Galois representations?

One aspect of anabelian geometry is the study of the action of the absolute Galois group of a field $K$ on the etale fundamental group $\pi_1(X_\overline K)$, where $X$ is a (anabelian) variety and ...

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### The Modularity Theorem and Serre's/Faltings's Isogeny Theorem

Earlier this year I completed my Masters dissertation on Andrew Wiles's proof of The Modularity Theorem for semistable elliptic curves as a precursor to the accepted proof of Fermat's Last Theorem. ...

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### Compact subgroups of linear groups over nonarchimedean fields

Let $n \in \mathbb{N}$, $K$ a (nonarchimedean) local field, $\overline{K}$ its algebraic closure. Take a compact subgroup $G \leq \text{GL}_n(\overline{K})$. Must there be a finite extension $F$ of ...

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### Faithful representations of free pro-p groups

Let $p$ be a prime number, $m,n \in \mathbb{N}$, $F = F(p,m)$ be the free pro-$p$ group on $m$ generators. For which $(m,n)$ there is a continuous faithful representation (embedding) $\rho : F ...

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### Cyclotomic character in class field theory

Let $K$ be an extension of $\mathbb{Q}_p$.
By local class field theory, the $p$-adic cyclotomic character $\mathrm{Gal}_K \rightarrow \mathbb{Z}_p^\times$ corresponds to a character $\chi : K^\times ...

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### If $G$ is compact, $H \leq G$ open, $V$ an irreducible $H$-rep, is $\text{Ind}_H^G$ semisimple?

Let $G$ be a compact group, $H$ a normal open subgroup, and $K$ a $p$-adic field (so that not all $G$-reps with coefficients in $K$ are semisimple). Let $V$ be a finite-dimensional topological ...

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### Connection of Galois representation and arithmetic geometry

This is might be a dumb question. There are lots of Galois representations which arise naturally from geometric objects, for example, Galois representations attached to elliptic curves. I know that ...

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### Bloch Kato Exponential as formal lie group exponential

Let $K$ be a $p$-adic field and $V$ a $p$-adic representation. In their paper on tamagawa numbers of motives, Bloch and Kato define an exponential map as the connecting homomorphism
$$DR(V) ...

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### About the restriction of a modular representation to a decomposition subgroup II

This question is a variant of this one.
Let $f$ be as in the other question, but suppose that we look at the $\ell$-adic representiation attached to $f$:
$$
\rho_f : G_{\mathbb Q} \to ...

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### About the restriction of a modular representation to a decomposition subgroup

Let $f$ be an eigenform of level $\Gamma_1(N)$ and let $p$ be a prime that does not divide $N$. It is well know that there is a $2$-dimensional representation
$$
\rho_f \colon G_{\mathbb Q} \to ...

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### 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
...

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### Newform and Galois representation (Shimura-Deligne Reciprocity Law)

Shimura-Deligne Reciprocity Law implies that to a newform $f \colon =f(z) \in S^{\mathrm{new}}_k(\Gamma_0(N))$ of weight $k$, one can associate Galois representation $\rho_{f,\lambda} \colon ...

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### Absolutely irreducible p-adic representation of the absolute Galois group of Q_p

Let $p$ be a prime number, $\mathbb{Q}_p$ the field of $p$-adic numbers, $G_p$ the absolute Galois group of $\mathbb{Q}_p$ and $V$ a finite dimensional vector space over $\mathbb{Q}_p$. Assume we are ...

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### Elliptic curve and Galois representation

For an elliptic curve $E$ over ${\Bbb{Q}}$, let us consider Serre's mod $l$ representation by
$\rho_{E,l} \colon {\mathrm{Gal}}({\overline{\Bbb{Q}}}/{\Bbb{Q}}) \to {\mathrm{Aut}}(\phantom{}_lE) = ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### Irreducibility of Galois representations attached to unitary groups

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 ...

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### Universal deformations of modular Galois representations

Let $\bar\rho$ be an odd, absolutely irreducible, 2-dimensional mod $p$ representation of $\operatorname{Gal}(\overline{\mathbf{Q}} / \mathbf{Q})$ (with coefficients in some finite extension $k / ...