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7
votes
1answer
234 views

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 ...
5
votes
1answer
276 views

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 ...
2
votes
0answers
90 views

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) ...
2
votes
1answer
119 views

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 ...
2
votes
1answer
161 views

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 ...
2
votes
2answers
348 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 ...
2
votes
1answer
164 views

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 ...
5
votes
2answers
217 views

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 ...
6
votes
1answer
343 views

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) = ...
9
votes
1answer
284 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 ...
10
votes
0answers
95 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 ...
5
votes
2answers
333 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 ...
3
votes
1answer
152 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 ...
8
votes
1answer
199 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 ...
8
votes
0answers
193 views

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 ...
8
votes
1answer
226 views

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 / ...
6
votes
0answers
122 views

classifying reducible 2-dimensional mod-p Galois representations

I want to classify reducible $2$-dimensional mod-$p$ Galois representations of a field $E$ of characteristic $p > 0$ (i.e. representations $G_E = \mathrm{Gal}(E^{sep}/E) \to GL_n(\mathbf{F}_p)$) $$ ...
14
votes
1answer
333 views

Local-global principle for split extensions of Galois representations

I guess the following is well-known (and probably follows from Chebotarev's density theorem, but I'm not very comfortable with it): Define some notation: $K$ a global field, $G$ the absolute Galois ...
1
vote
0answers
93 views

$(\varphi, \Gamma)$-modules, geometric interpretation $D_{diff}$

Could anyone explain to me the first paragraph of page 29 (IV.4.1) of this course of L. Berger: http://perso.ens-lyon.fr/laurent.berger/articles/article05.pdf Specifically, I would like to ...
3
votes
0answers
135 views

A question on a paper by Ribet

I'm reading the article On the equation $a^p + 2^\alpha b^p + c^p = 0$ by Ribet (http://math.berkeley.edu/~ribet/Articles/acta.pdf), but I'm having trouble understanding his proof of Theorem 3. For ...
5
votes
2answers
259 views

Reference for $p$-adic Hodge theory with coefficients

Let $K$ be a $p$-adic field and $L$ be a finite or infinite extension (maybe algebraic ?) of $\mathbb{Q}_p$. Is there a reference for $p$-Hodge theory for representations $\rho : Gal_K \rightarrow ...
4
votes
1answer
144 views

Properties of representations attached to p-adic modular forms

I found an old MOF post about representations attached to p-adic modular forms: Representations attached to p-adic modular forms and I have some follow up questions on the same topic. If we have a ...
4
votes
0answers
126 views

Is there a version of Serre's modularity conjecture for projective representations?

Serre's modularity conjecture asserts that a continuous odd irreducible representation $$\overline{\rho} : G_\mathbb{Q} \rightarrow \mathrm{GL}_2(\mathbb{F}_q)$$ must be modular, in the sense that ...
5
votes
1answer
134 views

Extending a representation from the Weil group to the Galois group

Let $F$ be a nonarchimedian local field. Since the Weil group $W_F$ is a dense subgroup of $G_F=Gal(\bar{F}/F)$, it's clear that restriction gives an injection $Irr(G_F)\rightarrow Irr(W_F)$ of ...
2
votes
0answers
56 views

Isometric representation semisimple?

The first lemma on p.35 of these notes states that unitary representations are semisimple. Could the same be said of isometries if the space doesn't have an inner product? This topic notes that the ...
6
votes
2answers
359 views

Misunderstanding in the hypotheses of Schlessinger's criterion

Good day to everyone. In studying deformation theory of Galois representations, I've come surely to an error, relating Schlessinger's criterion. Let's fix a representation $\bar{\rho}$ of a group ...
2
votes
1answer
163 views

When does the filtration in the limit of the Leray spectral sequence split?

Let $\ell$ be a prime, and $k$ a field of characteristic $\ne \ell$. Let $f \colon X \to Y$ be a proper map of smooth projective $k$-varieties. The Leray spectral sequence says $$ E_{2}^{pq} = ...
7
votes
1answer
355 views

Explicit calculation of Weil Deligne representations

According to Grothendieck monodromy theorem, l-adic galois representations of a local field corresponds to Weil-Deligne representations. However, given a galois representation, it is usually difficult ...
4
votes
2answers
327 views

Irreducible representations of compact groups

Let G be a compact group (or even profinite - Galois group). Let $V$ be a vector space over the field ${\mathbb F}_p$ with $p$ elements, $p$ a finite prime, such that $V$ is a contable product of ...
5
votes
0answers
145 views

When are Galois representations with open image attached to elliptic curves?

Let $K$ be a number field with absolute Galois group $G_K$. Let $\rho:G_K \rightarrow GL_2(\hat{\mathbb{Z}})$ be a Galois representation such that the image of $\rho$ is open in ...
3
votes
1answer
236 views

Are elliptic Kummer extensions big?

Loosely speaking, are elliptic Kummer extensions big? More concretely: Let $E$ be an elliptic curve over $\mathbb{Q}$, let $p$ be a prime, and let $F$ be a subfield of $\overline{\mathbb{Q}}$ ...
0
votes
0answers
84 views

Super-Gorenstein ideal of ${\Bbb F}_p[[X_1,\ldots,X_n]]$

Let $A \colon= {\Bbb F}_p[[X_1,\ldots,X_n]]$ be a $n$-variable power series ring over a finite field ${\Bbb F}_p$. We put ${\frak m}_A \colon= (X_1,\ldots,X_n)$. Definition(Super-Gorenstein ideal): ...
9
votes
0answers
383 views

Langlands program beyond CM fields?

I apologize since this is a quite vague question. And I am personally at an expert in these fields at all. It seems to me that there are two main directions of the Langlands program, namely, ...
5
votes
1answer
219 views

Psi operator on Phi-Gamma modules

This is a question about the base-rings appearing in the the theory of $(\varphi, \Gamma)$-modules in $p$-adic Hodge theory. Let $p$ be prime, $n \ge 1$, and let $$ ...
18
votes
0answers
556 views

Eichler-Shimura over Totally Real Fields

By Eichler-Shimura over totally real fields I mean the following conjecture. Conjecture. Let $K$ be a totally real field. Let $f$ be a Hilbert eigenform with rational eigenvalues, of parallel weight ...
6
votes
0answers
338 views

Benedict Gross's paper on companion forms

In the page 458 of his paper(A tameness criterion for Galois representations associated to modular forms), Gross wrote the following "A detailed analysis of $U_p(Af)+V_p(<p>f)$ shows that it ...
5
votes
1answer
139 views

Pro-$l$ Sylow action in a primitive representation of inertia over $\overline{\mathbb{F}}_l$

Let $K$ be a nonarchimedean local field of residue characteristic $p \neq l$ and let $I_K$ be the inertia subgroup of its absolute Galois group. Let $V$ an irreducible representation of $I_K$ over ...
5
votes
0answers
227 views

Which de Rham representations are trianguline?

Let $K/\mathbf{Q}_p$ be a finite extension, and let $V$ be an $n$-dimensional $\overline{\mathbf{Q}_p}$-vector space with a continuous action of $G_K$. Suppose $V$ is de Rham, so potentially ...
6
votes
0answers
240 views

Galois action on $E_n$-operads

Let $E_n$ be the little $n$-cubes operad which acts on $n$-fold loop spaces (up to group completion, an $E_n$-action is precisely the data needed to perform an $n$-fold delooping). I am looking for ...
6
votes
1answer
179 views

$(\varphi, \Gamma)$-modules of finite height

Maybe the answer to my question is obvious. Let $p$ be a prime $\geq 3$. Let $D$ be an étale $(\varphi, \Gamma)$-module over $A_{\mathbb{Q}_p} = \{ \sum_{n \in \mathbb{Z}} a_n X^n \, \vert \, a_n \in ...
10
votes
0answers
382 views

Why does $H^i(X_{ét},\mathbb{Q}_p)$ have a Hodge-Tate structure?

Let $X$ be a variety over a $p$-adic field $K$. Is there a simple or intuitive explanation of why the $G_K$ representation $H^i(X_{ét},\mathbb{Q}_p)$ is Hodge-Tate? More precisely, why do the powers ...
12
votes
1answer
308 views

What is the classification of characters in $p$-adic Hodge theory?

Let $K$ be a $p$-adic field and $\chi : Gal_K \rightarrow \mathbb{Q}_p^\times$ be a character. I know that $\chi$ is Hodge-Tate of weight $0$ iff $\chi(I_K)$ is finite (by Sen's theory), and that it ...
4
votes
1answer
188 views

$(\varphi, \Gamma)$-module of dimension 2 modulo $p$

Let $p$ be a prime number $\geq 3$. Let $V$ be a representation of $Gal(\bar{\mathbb{Q}}_p/ \mathbb{Q}_p)$ with coefficients in $\mathbb{F}_p$. Assume $V$ is a non-split extension of two characters ...
6
votes
2answers
583 views

Peu or très ramifiée extension

Let $p$ be a prime number. Let $\mathbb{F}$ be a finite extension of $\mathbb{F}_p$. Let $\omega$ be the mod $p$ cyclotomic character and let $V$ be a representation of $G_{p} = Gal(\bar{\mathbb{Q}}_p ...
4
votes
2answers
218 views

Generalization of Kummer isomorphism?

This is a question I asked on math.stackexchange without success. Let $p$ be a prime number and denote by $\mathbb{F}_p(1)$ the one dimensional vector space over $\mathbb{F}_p$ endowed with an action ...
3
votes
1answer
222 views

$\ell$-conductor of a two-dimensional $\ell$-adic Galois representation

Let $\ell$ be a prime number, denote by $K_\ell$ the maximal algebraic extension of $\Bbb{Q}$ ramified only at $\ell$. Let $f = \sum a_n q^n$ be a Hecke eigenform of level $1$ with integer ...
2
votes
0answers
130 views

Local components of quaternionic modular forms

Let $D$ be a totally definite quaternion algebra over a totally real number field $F$. Let $U$ be an open compact subgroup of $D(\mathbb{A}_F)^\times$, maximal compact almost everywhere. Consider ...
8
votes
1answer
312 views

Extensions of Galois representations

Let $G=Gal(\bar{\mathbb Q}/{\mathbb Q})$ be the absolute Galois group of the rationals. Fix two continuous group homomorphisms $\alpha,\beta: G\to {\mathbb Q}_l^\times$, where $l$ is a prime and ...
2
votes
1answer
228 views

Galois deformations with Panchiskin condition

Let $L/\mathbf{Q}_p$ be a finite extension and we consider a fixed $L$-linear representation $V$ of the absolute Galois group $G:=\operatorname{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$. Assume that ...
2
votes
0answers
115 views

Residue fields of attached to coefficients of modular forms

Let $f = \sum_n a_n q^n$ be a cuspidal newform of some weight and level. Here I want to view the $a_n$ of $p$-adic numbers (by embedding $\overline{{\bf Q}}_p$ in ${\bf C}$ in some way). Let ...