Questions tagged [galois-representations]

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.

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0 answers
138 views

Power series ring $R[[X_1,\ldots,X_d]]$ over a domain $R$

Let $R$ be a domain and \begin{align*} T \,\colon= R[[X_1,\ldots,X_d]]. \end{align*} Suppose that we have $d$ elements $f_1,\ldots,f_d \in T$ and let us consider an ideal $J$ of $T$ such that $(f_1,\...
2 votes
1 answer
242 views

Étale fibration for $K[[X_1,...,X_n]]$

Let us consider a formal power series ring $A_n \colon= K[[X_1,\ldots,X_n]]$ with $0 \ll n < \infty$ and we shall consider a prime ideal ${\frak P}$ of $A_n$ such that $1 < {\mathrm{ht}}({\frak ...
6 votes
1 answer
639 views

Connection of Galois representation and arithmetic geometry

There are lots of Galois representations which arise naturally from geometric objects, for example, Galois representations attached to elliptic curves. I know that people are interested in studying ...
2 votes
0 answers
106 views

Deforming Modular Symbols

This is probably a silly question, as I don't really know a whole lot about modular symbols over arbitrary rings. How do modular symbols over a finite field square with Katz modular forms? If they ...
1 vote
0 answers
288 views

Primes of the power series rings

Let $A_n \colon= K[[X_1,\ldots,X_n]]$ be a $n$-variable formal power series ring. By setting $X_n \mapsto 0$, we obtain a natural surjection \begin{equation*} \psi_{n,n-1} \colon A_n \...
2 votes
1 answer
262 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 \...
0 votes
0 answers
116 views

Gauss lemma for a complete Noetherian domain

Suppose that $R$ is a Noetherian complete domain over a field $K$. Suppose that a monic polynomial $f(X) \in R[X]$ (i.e., the highest degree $X^e$ in $f$ has the coefficient $1$), satisfies the ...
5 votes
0 answers
295 views

Hodge-Tate weights of cohomological cuspidal automorphic representation

Let $\Pi$ be an algebraic cuspidal automorphic representation for $GL_{n}/\mathbb{Q}$ cohomological with respect to a dominant integral weight $\mu \in X^{*}(T)$ ($T \subset GL_{n}$ being the standard ...
1 vote
0 answers
118 views

On the exponent of a certain matrix $A$ in characteristic $p > 0$

Let $A$ be a square matrix in characteristic $p > 0$ with both column and row having length $(1 + p^0 + p + \cdots + p^i)$, where $i \geq 0$. Suppose that further the $(m,n)$-component $a_{m,n}$ ...
5 votes
1 answer
436 views

Nearby cycles and extension by zero

Let $f: X\to \text{Spec}(R)$ be a proper and smooth morphism, with $R$ a strictly henselian dvr. Call $s = \overline{s}$ the closed point and $\eta$ the geometric point of $\text{Spec}(R)$. Call $i_s ...
2 votes
0 answers
226 views

Berthelot’s comparison theorem and functoriality

Let $A$ be a noetherian $p$-adically complete ring with an ideal $I$ equipped with a PD structure and such that $p$ is nilpotent on $A/I$. Let $S = \text{Spec}(A)$, $S_0 = \text{Spec}(A/I)$, $Y\to S$ ...
2 votes
0 answers
154 views

Classification of mod p Galois Representations for l not equal to p

Let $l\neq p$ be primes and let $\text{G}_l:=\text{G}_{\mathbb{Q}_l}$. Let $k$ be a finite field of characteristic $p$ and $\bar{\rho}:\text{G}_l\rightarrow \text{GL}_2(k)$ a local Galois ...
2 votes
0 answers
132 views

Iwasawa Theoretic Interest in a certain type of result

This question is probably going to sound vague (since it does to me) and I wish I could make it more precise, but here goes. For $p\in \{107, 139, 271,379\}$ Ohtani and Blondeau (in separate papers) ...
15 votes
1 answer
1k views

Some basic questions on crystalline cohomology

Let $X_0$ be a smooth projective variety over $\mathbf{F}_q$ and ${X}$ its base change to an algebraic closure $k$ of $\mathbf{F}_q$. Crystalline cohomology $H^*_{\rm cris}(X) := H^*((X/W(k))_{\rm ...
5 votes
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171 views

Is there a conjectural analog of Ribet's theorem (Converse to Herbrand's Theorem) for Imaginary Quadratic fields?

For $p$ a prime, let $Cl(\mathbb{Q}(\mu_p))$ denote the class group of the extension of $\mathbb{Q}$ obtained by adjoining a primitive $p$th root of unity. Associated to an eigenform of weight 2 and ...
6 votes
1 answer
502 views

How to compute Galois representations from etale cohomology groups of a generalized flag variety?

Let $G$ be a connected reductive group over a number field $K$, $P$ be a parabolic subgroup of $G$ defined over $K$, $X=G/P$ be the generalized flag variety which is a smooth projective variety over $...
3 votes
0 answers
472 views

Condition on a Fontaine Laffaille module which prescribes the image of the associated Galois representation

The Setup: Let $m\geq 1$ be an integer, $\mathbb{F}$ be a finite field of characteristic $p$ and $W(\mathbb{F})$ the ring of Witt-vectors with residue field $\mathbb{F}$ and $\sigma:W(\mathbb{F})\...
7 votes
0 answers
369 views

Local properties of Galois representations attached to torsion classes

$\DeclareMathOperator{\PGL}{PGL} \DeclareMathOperator{\GL}{GL} \newcommand{\F}{\mathbb{F}} \newcommand{\p}{\mathfrak{p}} \DeclareMathOperator{\Sym}{Sym}$ Let $F$ be a number field, and let $\Gamma$ be ...
9 votes
1 answer
495 views

A question about Galois representations

Let $K$ be a number field and $(\rho,V)$, $(\rho',V')$ be two Galois representations of $\mathrm{Gal}(\overline{\mathbb{Q}}/K)$. Suppose that for some positive integer $n$ we have $\mathrm{Sym}^n\rho\...
6 votes
1 answer
502 views

Functoriality for $\ell$-adic cohomology - a question

This should a be basic enough question, but I’m a little confused. In proving that $H^*(X,\mathbf{Q}_{\ell})$ is functorial (in the sense of Weil cohomology theories: see axiom D2 here) as $X$ ranges ...
6 votes
1 answer
530 views

Quaternion algebra actions on $\ell$-adic cohomology

Let $E$ be a supersingular elliptic curve over $\mathbf{F}_p$, and $H$ its endomorphism algebra $\text{End}(E)\otimes_{\mathbf{Z}}\mathbf{Q}$, a quaternion algebra (non split at $p$ and $\infty$). ...
2 votes
0 answers
130 views

Local polynomials of Frobenius-semisimple Weil representations which are tensor products of an Artin representation and an unramified character

Let $K$ be a local field and $\rho: W_K \to \operatorname{GL}(V)$ be a Weil representation. The for any finite extension $F/K$, we define the local polynomial $$ P(\rho|_F,T) = \det{(1 - \operatorname{...
11 votes
1 answer
631 views

Finiteness or infiniteness for Galois representations with unusual Hodge numbers

Say a representation $\operatorname{Gal}(\mathbb Q) \to GL_n(\overline{\mathbb Q}_\ell)$ has big monodromy if the Zariski closure of the image of $\operatorname{Gal}(\mathbb Q) $ contains $SO_n$ or $...
5 votes
0 answers
195 views

Is there some computational evidence of the $pq$ analog of Serre's conjecture?

The $pq$ analog of Serre's conjecture (see "Mod pq Galois representations and Serre's Conjecture"- Khare, Kiming) states that if $\bar{\rho}_1:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{F}_p)$ is a ...
2 votes
0 answers
146 views

Central character of automorphic representations of $Sp_{2n}$

Let $F$ be a CM field. Given a regular algebraic self-dual cuspidal automorphic representation $\Pi$ of $GL_n(\mathbb A_F)$ and a prime $l$, there is a continuous Galois representation $r_{\Pi}: \...
19 votes
0 answers
2k views

A question in Fontaine--Laffaille theory

Let $K$ be finite unramified extension of $\mathbf{Q}_p$ with ring of integers $W$. Let ${\rm MF}$ be the category of strongly divisible $W$-modules $M$ with ${\rm Fil}^0M=M$ and ${\rm Fil}^{p-1}M=0$. ...
3 votes
1 answer
366 views

Is there relationship between $\mu=0$ for an elliptic curve and the irreducibility of its residual representation at a prime $p$?

I remember it being mentioned at a talk that if $E$ is an elliptic curve over $\mathbb{Q}$ and $p$ a prime at which it has good reduction then the dual to the Selmer group over the cyclotomic $\mathbb{...
2 votes
1 answer
291 views

Heights of contracted ideals

Let $R$ be a non-noetherian domain. $S$ be a multiplicatively closed subset of $R$. Let $S^{-1}R$ be a localisation of $R$, where all element of $S$ is invertible. Suppose we have an ideal $I$ of $R$. ...
3 votes
1 answer
473 views

Conductor of Galois representation attached to newform

(Sorry for poor my english skill..) Let $k$ and $N$ be positive integers and $\chi$ be a Dirichlet character modulo $N$. Let $F$ be a newform with number field $K_{F}$. (All coefficients of $F$ in $...
7 votes
0 answers
293 views

List of applications of the Deformation theory of Galois Representations in contexts other that $R=T$

The Deformation theory of Galois representations developed by Mazur has been extensively used in proving that Galois representations with special properties are modular/automorphic through the ...
5 votes
1 answer
355 views

Is the following variant of Shafarevich's theorem known?

Let $Q$ be a finite simple group which may be realized as the Galois group of some extension of $\mathbb{Q}$ (like for instance $PSL_2(\mathbb{F}_p)$ for $p\geq 5$, or the monster group) and let $G$ ...
2 votes
1 answer
370 views

Classes of hyperplane sections in cohomology

Let $X$ be a smooth projective variety over the algebraic closure of a finite field with Galois group $G$. Is it true that the vector space $H^{2k}(X,\mathbf{Q}_{\ell}(k))^G$ has always positive ...
5 votes
1 answer
465 views

Frobenius eigenvalues algebraic numbers

Let $X$ be a smooth projective variety over $\mathbf{F}_q$ and $\overline{X}$ its base change to $\overline{\mathbf{F}_q}$. By Deligne’s Weil I, the eigenvalues of the geometric Frobenius acting on $...
6 votes
0 answers
134 views

Congruence between modular forms

This might be a very vague question since I am not very familiar with the theory of automorphic forms. Let $G$ be a connected reductive algebraic group defined over $F$ (a number field). Suppose we ...
4 votes
2 answers
438 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 ...
4 votes
2 answers
822 views

Serre's conjecture for mod-p^n representations?

I think this may be a silly question, but here goes. Let $\rho:\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to \mathrm{GL}_2(\overline{\mathbf{F}_p})$ be a representation; say $\rho$ is of S-type ...
23 votes
2 answers
2k views

Even Galois representations "mod p"

Consider an irreducible $\mathrm{mod}$ $p$ representation: $$\rho: \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_2(\bar{\mathbb{F}}_p)$$ If $\rho$ is odd, it was conjectured by Serre in ...
3 votes
1 answer
130 views

If $\Pi$ and $\Sigma$ agree at almost all places, then the central character of $\Pi$ corresponds to $\operatorname{Det} \Sigma$

Let $\Sigma$ be an $n$-dimensional representation of the global Weil group $W_F$ for a number field $F$, and $\Pi$ an automorphic representation of $\operatorname{GL}_n(\mathbb A_F)$. Suppose that at ...
7 votes
1 answer
427 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 $GL_2(\hat{\mathbb{...
3 votes
1 answer
169 views

Congruence of normalized eigenforms at two primes

Let $f_i\in S_{k_i}(\Gamma_0(N_i))$ be normalized cuspidal eigenforms for $i=1,2$ and let $K$ be the composite of the fields of Fourier coefficients generated by $f_1$ and $f_2$ and let $\mathfrak{p}...
2 votes
0 answers
137 views

Hodge-Tate weights of etale cohomology groups

Given a smooth algebraic variety $X$ over a number field $F$, its $p$-adic cohomology groups $H^i(X \times_F \bar F, \mathbb Q_p)$ carries an action of $\mathrm{Gal}(\bar F/F)$, which gives a ...
6 votes
1 answer
187 views

Is it expected that the mod $p$ representation determines a normalized Hecke newform of fixed weight for p large enough?

Mazur's conjecture on the image of Galois representations of Elliptic curves states that for $N$ large enough there is a unique elliptic curve $E$ over $\mathbb{Q}$ giving rise to a fixed mod $N$ ...
1 vote
0 answers
102 views

Local factors determine Weil representations - proof of the Artin representation case

This post can be seen as a continuation of this post I created on MathOverflow. I want to understand the proof of the following Theorem from "Euler Factors determine Weil Representations" by Tim and ...
4 votes
1 answer
253 views

Are there Galois representations associated with any regular algebraic cuspidal automorphic representation?

Let $K$ be a number field and $\pi$ be a regular algebraic cuspidal automorphic representation on $\mathrm{GL}_2(\mathbb{A}_K)$. Let $\lambda$ be a prime of the field of Fourier coefficients of $\pi$ ...
2 votes
1 answer
138 views

Steinberg components of local deformation rings

Let $F=\mathbb Q_l$ and $E$ be a finite extension of $\mathbb Q_p$ with residue field $k \cong O_E/m_E$ ($p \neq l$). Let $\bar r: \Gamma_F=Gal(\bar F/F) \to GL_2(k)$ be a continuous representation. ...
2 votes
0 answers
161 views

Prime ideals being finitely-generated implies coherence?

Let $R$ be a non-noetherian local domain. Suppose that the following two conditions hold for $R$$\colon$ $(*)$$~\quad$An arbitrary prime ideal ${\frak P}$ of $R$ such that ${\mathrm{ht}}({\frak P}) &...
2 votes
1 answer
281 views

Local factors determine Weil representations - proof of the cyclic case

I already created this post on Math Stack Exchange but I was not so sure if this question fits better here. If it is not, I want to apologize in advance and feel free to delete my post. I want to ...
7 votes
2 answers
906 views

Is every 3-dim self-dual Galois representation a symmetic square of 2-dim representation?

Basically, my question as in the title. Here the Galois representation I consider is an $\ell$-adic Galois representation (comes from geometry). And by the word "self-dual" I mean that representation ...
6 votes
0 answers
233 views

Galois actions on cohomology rings of algebraic varieties

Let $k$ be an arithmetic field. Let $G_k$ be its absolute Galois group. $G_k$ is often studied via its linear action on cohomology (etale, crystalline, ...) "groups" of algebraic varieties over $k$. ...
6 votes
0 answers
321 views

Embeddings of number fields into $\mathbb{C}$ or $\bar{\mathbb{Q}}_l$

When studying arithmetic Galois representations for a number field $F$ one often fixes at the outset an embedding of its algebraic closure $\bar{F}$ into $\mathbb{C}$ or $\bar{\mathbb{Q}}_l$ and ...

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