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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Decomposition of the Galois group of the $m$-th division field of an elliptic curve with CM into a direct product of Galois groups

Let $E/\mathbb{Q}$ be an elliptic curve with CM from an imaginary quadratic field $K$. Let $K(E[m])$ denote $m$-th division field (number field obtained by adjoining the coordinates of the $m$-torsion ...
Anish Ray's user avatar
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How does the cohomology of the Lubin-Tate/Drinfeld tower fit into categorical p-adic local Langlands?

In conjecture 6.1.14 of this article, Emerton-Gee-Hellmann formulate the p-adic local Langlands conjecture, which posits the existence of a fully faithful functor from (the appropriate derived ...
Anton Hilado's user avatar
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Galois representations with semisimple residue representation

$\DeclareMathOperator\GL{GL}$Let $\mathbb{Z}_p$ be the ring of integers of $p$-adic numbers $\mathbb{Q}_p$, $G$ a profinite group (e.g. Galois group of local field or global field) and $\rho:G\to \...
stupid boy's user avatar
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Why does Deligne's construction of the Galois representation attached to the new cuspidal forms require that the Kuga-Sato manifold be regular?

The origin of this question is related to the construction of Galois representations of Deligne attached to $f$ a new cuspidal form (of weight $k\geq 2$). To do this, we consider the fiber product $k$-...
Marsault Chabat's user avatar
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A question on a paper of K. Murty

Let $f=\sum_{n\ge 1}a_nq^n$ be a normalized Hecke eigenform which is not of CM-type, of weight $k\ge 2$ for the congruence subgroup $\Gamma_0(N)$. Let $a\in\mathbb{Z}$ and define $$ \pi_f(x,a):=\#\{p\...
Fouad Fahmi's user avatar
6 votes
0 answers
364 views

How to construct this non-geometric mod $p$ Galois representation?

Let $ K $ be a totally real cubic field. I am concerned with the following kind of Inverse Galois Problem: Is there a prime $ p>5 $ and a continuous representation $ \bar{\rho}:G_{K}\to {\rm GL}_{...
Nobody's user avatar
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Infinitely ramified $p$-adic representations as limits of finitely ramified $p$-adic representations

Let $ \rho:G_{\mathbb{Q}}\to {\rm GL}_{2}(\mathbb{Q}_{p}) $ be a continuous $ p $-adic representation of the absolute Galois group $ G_{\mathbb{Q}} $ of the rational number field $ \mathbb{Q} $. Let $ ...
Nobody's user avatar
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Purity for proper varieties

Let $X$ be a proper, geometrically connected, geometrically integral variety over $\mathbf{F}_q$. There exists a finite field extension $k/\mathbf{F}_q$ of degree $d$ and an alteration $X'\to X_k$ ...
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In practice, how explicitly can we describe a Galois representation?

In most parts of representation theory, we naturally want to describe a given representation as explicitly as possible. This seems to me a feasible project only if, as a first step, the group itself ...
Kim's user avatar
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Crystalline when restricted to inertial subgroup

$\newcommand{\ur}{\mathrm{ur}}\newcommand{\cris}{\mathrm{cris}}$Let $K$ be a finite extension of $\mathbb{Q}_p$, $G_K=\operatorname{Gal}(\overline{K}/K)$ and $I_K \subset G_K$ its inertial subgroup. ...
Desunkid's user avatar
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Trianguline representation

I have a problem in understanding the concept of trianguline representation. Maybe someone can enlighten me. Let $K$ be a finite extension of $\mathbb{Q}_p$ and $V$ be a $p$-adic representation of $...
Konstantin's user avatar
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Status of Mazur's dimension conjecture on universal deformation rings

Fix a prime $ p $. Let $ \mathbb{F} $ be a finite field of characteristic $ p $ and $ W(\mathbb{F}) $ the ring of Witt vectors of $\mathbb{F} $ . Let $ \widehat{\mathfrak{A}}_{W(\mathbb{F})} $ be the ...
Nobody's user avatar
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Existence of even mod $ p $ Galois representations with full image

My question is that: For any prime $p>7$, is there a mod $ p $ representation $ \bar{\rho}:G_{\mathbb{Q}}\to \operatorname{GL}_{2}(\mathbb{F}_{q}) $ of the absolute Galois group $ G_{\mathbb{Q}} $ ...
Nobody's user avatar
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On presentations of universal rings of deformations

Let $k$ be a finite field of characteristic $p$, and $R$ a complete local noetherian algebra with residue field $k$. It is well known that $R$ has a natural structure of an algebra over the ring Witt ...
Fernando Peña Vázquez's user avatar
3 votes
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Galois invariant of Tate module

Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $V$ be a de Rham representation of $G_K=\operatorname{Gal}(\overline{K}/K)$. By Corollary 3.8.4 of Bloch and Kato - L-functions and Tamagawa ...
Desunkid's user avatar
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Image of the Kummer map for abelian varieties over $p$-adic local fields

The following statement might be well-known to the community: let $K$ be a finite extension of $\mathbb{Q}_p$ for some prime $p$. Let $A$ be an abelian variety over $K$. Then the image of the Kummer ...
User0829's user avatar
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On the pro-category of finite local artinian algebras

Let $\mathbb{F}$ be a finite field, and $W(\mathbb{F})$ its associated ring of Witt's vectors. On page 6 of the following lecture notes Deformations of Galois Representations, the category $\mathfrak{...
Fernando Peña Vázquez's user avatar
4 votes
1 answer
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Reductive subgroups of $\mathrm{GL}_2$ over an algebraically closed field of characteristic zero

I am reading a very nice paper of Newton and Thorne, Symmetric power functoriality for holomorphic modular forms, and there is an argument concerning the (Zariski-closure of) image of certain $p$-adic ...
babu_babu's user avatar
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Do we expect the Langlands correspondence to be a functor?

In the literature I've read, it is often said that to a Hecke eigenclass, one would like (and sometimes succeeds) to "associate" or show the existence of a Galois representation such that ...
curious math guy's user avatar
1 vote
1 answer
263 views

Explicit Chebotarev density theorem for Galois representations associated to newforms

Let $f \in S_2(\Gamma_0(N))$ be a newform with associated residual Galois representation $\rho: \operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \operatorname{GL}_2(\mathbf{F})$, $\mathbf{F}$ ...
user471019's user avatar
5 votes
2 answers
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Irreducibility of the $n$th symetric power of the reduction of the Galois representation of a non-CM newform

In "On $\ell$-adic representations attached to modular forms II", Ribet proved that the $\ell$-adic representation $\rho_{f,\ell}$ attached to a non-CM newform form $f$ satisfies $${\rm SL}...
Fouad Fahmi's user avatar
3 votes
1 answer
402 views

Galois cohomology of abelian varieties

Suppose $A$ is an abelian variety over a number field $K$ and call $M$ the maximal torsion free quotient of $A(\overline{K})$ equipped with its Galois action. For the first Galois cohomology of $M$, ...
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Galois extension of $ C_{k} $ field and irreducible polynomial over $ C_{k} $ field

A field is called a $ C_{k}$- field if every form i.e. every homogeneous polynomial of degree $ d $ in $ n > d^{k} $ indeterminates has a nontrivial zero. We know that any finite field is a $ C_{...
Sky's user avatar
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Constructing motivic representations through extensions of $\mathrm{SL}(2, \mathbb{Z})$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$In the Esquisse, Grothendieck explains how to find the representations of $G_{\mathbb{Q}}$ on Tate modules of Jacobians through its action on $\...
user149000's user avatar
3 votes
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59 views

Family of residual representations compatible with a motive

Suppose $S$ is a set of primes. For each prime $p$ in $S$ we are given a Galois representation $\rho_p : Gal(\bar{\mathbb{Q}}/\mathbb{Q}) \rightarrow GL_2 (\mathbb{F}_p) $. What are the necessary or ...
Ghy's user avatar
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2 votes
1 answer
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Galois representations attached to a cusp form for different primes

If I have a cusp form $f$, I can consider the associated Galois representation $\rho_l(f)$ for any prime $l$. For two distinct primes $p$ and $q$, what is the relationship between $\rho_p(f)$ and $\...
user390304's user avatar
3 votes
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The cycle class map with values in crystalline cohomology

Let $ k = \mathbb{F}_q $ be a finite field of characteristic $ p > 0 $. Let $ X $ be a smooth proper scheme of dimension $ d $ over $ k $. Consider the associated $ K $ - linear cycle class map ...
Bradley04's user avatar
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9 votes
1 answer
294 views

A question about mod $p$ local Langlands for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$

In the mod $p$ local Langlands correspondence for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$, the irreducible supercuspidal representation $\left(\mathrm{ind}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}_{\mathrm{GL}_{2}(...
Mihir Sheth's user avatar
1 vote
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170 views

Compute dimension of space of modular forms by counting Galois representations

It is known that we can compute the dimension of the space $S_{k}^{\mathrm{new}}(N, \chi)$ of new forms of weight $k\geq 2$ and level $N$ and Nebentypus $\chi$ via Riemann-Roch theorem or using ...
Seewoo Lee's user avatar
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21 votes
1 answer
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Cohomology of Shimura varieties and coherent sheaves on the stack of Langlands parameters

In Zhu's Coherent sheaves on the stack of Langlands parameters theorem 4.7.1 relates the cohomology of the moduli stack of shtukas to global sections of a certain sheaf on the stack of global ...
Anton Hilado's user avatar
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3 votes
0 answers
276 views

Eichler orders in a certain quaternion algebra

Let us consider a totally real number field $K$ such that $[K \colon {\Bbb Q}] = {\mathrm{odd}}$. We shall consider the quaternion algebra $D$ over $K$ such that $D$ splits everywhere at finite places ...
Pierre MATSUMI's user avatar
8 votes
2 answers
531 views

Motivation of the construction of $p$-adic period rings

Let $B$ be either $B_{\text{dR}}$ or $B_{\text{crys}}$. For a $\mathbb{Q}_p$-representation $V$ of the absolute Galois group $\mathrm{Gal}(\overline{K}/K)$ of a $p$-adic field $K$ (a finite extension ...
User0829's user avatar
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3 votes
0 answers
201 views

How the Galois group acts on a Néron–Severi group of a variety?

Let $K/k$ be a Galois extension with Galois group $\Gamma$ and let $X$ be a variety over $k$. Assume that either $X(k)\neq\varnothing$ or $\mathrm{Br}(k)=0$, the Brauer group of $k$. By the Hochschild-...
Mobius's user avatar
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Determining existence of a $p$-isogeny from $p|E(\mathbb{F}_{\ell})$

In Siksek's notes The modular approach to Diophantine equations he uses the following result: Let $p$ be an odd prime. For an elliptic curve $E$ over $\mathbb{Q}$ if $p|E(\mathbb{F}_{\ell})$ then for ...
Μάρκος Καραμέρης's user avatar
3 votes
0 answers
105 views

A uniform version of Bashmakov's theorem for elliptic curves

Let $E/\mathbb Q$ be an elliptic curve. Serre's open image theorem is the statement that the image of the Galois group $G_{\mathbb Q}$ into $GL_2(\mathbb Z/n\mathbb Z)$ by it's action on the torsion ...
Asvin's user avatar
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2 votes
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191 views

Galois-action on spectral sequence

Let $X_\bullet\to S$ be a proper surjective hypercover of a $k$-scheme by smooth proper $k$-schemes. This gives a proper surjective hypercover $X'_\bullet\to S_{\bar{k}}$ where $X'_n:=X_n\times_k \bar{...
curious math guy's user avatar
3 votes
0 answers
146 views

$p$-adic Hodge theoretic properties of global Galois representations via $\ell$-Frobenii

Let $G_{\mathbb{Q},S} = \mathrm{Gal}(\mathbb{Q}_S/\mathbb Q)$ where $\mathbb Q_S$ is the largest algebraic extension of $\mathbb Q$ unramified outside a finite set of places $S$. Then the union over $\...
Ashwin Iyengar's user avatar
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75 views

An invariant subspace under $G_Q$ action , and BSD-rank

Let $E/Q$ be an elliptic curve. $E(\bar{Q})$ is a complicated abelian group, which equals to all closed points of $\bar{E}$, and also, a $G_Q$-Galois module. Its torsion part, $E(\bar{Q})_{tor}$ is a ...
Yuan Yang's user avatar
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Question on GSpin-Valued L-parameters

Let $\Gamma$ be a topological group, $n \geq 1$ an integer, $\ell$ a prime number, and $\overline{\mathbb{Q}}_{\ell}$ the algebraic closure of the $\ell$-adic integers. We set $\Phi(GSpin_{2n + 1})$ ...
DimDan's user avatar
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Reference Request - New proof of Ribet's level lowering by Khare and Wintenberger

I'm currently following the note of Sug Woo Shin's course at Berkeley with notes taken by Rong Zhou. In Section 24.3 (Page 86), Ribet's level lowering theorem is stated: [Theorem 24.7] $E = E_{a^{\...
Hetong Xu's user avatar
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Diophantine consequences of the Buzzard–Diamond–Jarvis conjecture

Serre's modularity conjecture famously implies Fermat Last Theorem. More generally, Serre's conjecture implies that certain generalized Fermat equations have no non-trivial solutions (see Section 4.1 ...
Tristan Phillips's user avatar
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A question on the Robba ring

Notation is as in the question: https://math.stackexchange.com/questions/4090045/some-questions-about-the-robba-ring. We define a new operator over the Robba ring as follows. Put $$c=\frac{pE(u)}{E(0)}...
Luc's user avatar
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0 answers
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Looking for a crystalline analogue of , $\mathcal {Z}_{\sim}^* (X)_F \simeq \mathcal{Z}_{\sim}^* (X_{k ' })_F^{\mathrm{Gal} ( k ' / k )} $

Is there a crystalline analogue for the following formula, using the crystalline Frobenius $ F_v $ instead of the absolute Galois group $ \mathrm{Gal} (\overline{k} / k) $ ? Here is the formula, which ...
Bradley04's user avatar
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Commutative algebra details on patching when proving $R = \mathbb{T}$ theorem (Calegari-Geraghty Paper)

I have originally posted this on math.SE and been suggested to post this here. I'm merely an undergraduate student and it is the first time for me to ask questions here. I'm sincerely sorry if these ...
Hetong Xu's user avatar
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4 votes
1 answer
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Meaning of Atkin-Lehner eigenvalues

Suppose I have $f\in S_2(\Gamma_0(N))$ a classical modular newform of level $N$. I want to understand what information (if any) is carried by its Atkin-Lehner eigenvalues for primes $p\mid N$, as ...
ribbcastle's user avatar
4 votes
1 answer
242 views

Semisimplicity of the étale cohomology mod $p$

Let $X$ be a smooth projective variety over a field $k$. Then if $\ell\neq \text{char} k$, $k$ is finite, and $X$ is an abelian variety it was shown by Weil that the $\ell$-adic cohomology of $X_{k^{...
curious math guy's user avatar
2 votes
1 answer
213 views

Local to global for semistable $G_{\mathbb{Q}_p}$-representations

Let $\rho_p:G_{\mathbb{Q}_p} \to \text{Gl}_n(\mathbb{Q}_p)$ be semistable representation. In local to global Galois representation, it was asked if one can find a geometric global Galois ...
curious math guy's user avatar
3 votes
1 answer
178 views

Restriction of $(\varphi, N)$-modules

For any $p$-dic field $K$, we have an equivalence of categories $$D_{st}:Rep_{\mathbb{Q}_p}^{st}(G_K)\rightarrow MF_K^{ad}(\varphi,N),\quad V\mapsto (B_{st}\otimes_{\mathbb{Q}_p} V)^{G_K}$$ with quasi-...
curious math guy's user avatar
7 votes
0 answers
434 views

Status of the conjectured vanishing of Bloch-Kato H^2

There is a folklore conjecture that $\operatorname{Ext}^2$ vanishes in the category of geometric $p$-adic Galois representations (i.e. representations that are unramified almost everywhere and de Rham ...
David Corwin's user avatar
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12 votes
1 answer
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Eigenvarieties and functoriality

In Langlands' review of Hida's book "$p$-adic automorphic forms on Shimura varieties", he discusses a nexus of 4 areas of modern number theory: automorphic representations, motives, spaces ...
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