Questions tagged [galois-representations]

Questions relating to the representations of the absolute Galois group $\mathrm{Gal}(\overline K/K)$ of a number field or of a local field.

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Galois representation on elliptic curves mod p

Let $E/K$ be an elliptic curve over number field $K$. Assume that the Galois representation $\rho_K : G_K \rightarrow GL_2(\mathbb{Z}/p\mathbb{Z})$ is surjective. Then we have the isomorphism $G(K(E[p]...
WHERE 234's user avatar
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Why is the subgroup of the adelic image $E_{/\mathbb{Q}}$ open?

I am reading Serre's 1972 paper (English) Proposition 22 (24 in English version) where he proves that no elliptic curve over $\mathbb{Q}$ can have surjective adelic image. In the proof he says This ...
Batrachotoxin's user avatar
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Road map to learning about $\ell$-adic representations of $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$

I am interested in learning about the congruence relations satisfied by the coefficients of modular forms, in particular I am interested in learning more about $\tau(n)$, the coefficients of the ...
Takamoto Yuji's user avatar
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How to understand Galois Representations as an undergraduate? [closed]

I am looking to doing my undergraduate thesis on Galois Representation Theory. I am currently taking a class in Galois Theory and learning Algebraic Geometry independently. Are there any resources ...
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Serre's definition of $U_{\mathfrak{m}}$ for $\ell$-adic representations

In Serre's "Abelian $\ell$-adic representations and Elliptic curves", in order to define the set $S_{\mathfrak{m}}$ he defines of $U_{v,\mathfrak{m}}$ as follows: $$U_{v,\mathfrak{m}}=\left\{...
Yang Awotwi's user avatar
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Ramification of mod $\ell$ representation of elliptic curves

Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $p,\ell$ be two prime numbers. Consider the mod $\ell$ representation $$\rho:Gal(\mathbb{\overline{Q}}/\mathbb{Q})\to Aut(E[\ell])= GL(2,\ell).$$...
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$ \mathbb{Q}_p $-admissible p-adic representaion

In Serin Hong's notes on $p$-adic hodge theory, he claimed that every $p$-adic representation is $\mathbb{Q}_p$-admissible, as $D_{\mathbb{Q}_p}$ is the identity functor.But by the definition of the $...
Kevin's user avatar
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Regular representation of the Galois groups

Let $G_\overline{\mathbb Q}$ the absolute Galois group of $\mathbb Q$ and let $G'$ the subgroup of order $2$ generated by complex conjugation. Now, I've show that $G_\overline{\mathbb Q}/G'$ is a ...
Luis Antonio Sanchez's user avatar
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Definition of Artin $L$-functions

I am reading notes of Samuel Marks on ”Galois Representations”. In this paper, on page 14 (definition 3.1), it is said that given a representation $\rho$ of $Gal(L/K)$ it yields another representation ...
confused's user avatar
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Discrete group action and openness of stabilizer

Let $G$ be a topological group that acts (topologically) on a topological space $V=\mathbb Q_p^n$. Moreover, let us assume that this action is linear, in other words, the action gives a linear ...
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$Gal(LM/K)$ isomorphic to $ \{ (\sigma , \tau) \in Gal(L/K) \times Gal(M/K) : \sigma |_{L \cap M} = \tau |_{L \cap M} \} $

Let L/K, M/K be finite Galois extensions contained in some common field extension $\mathbb{K}$ of $K$, so we can speak about $L \cap M$ and the composite $LM$ , which is defined to be the smallest ...
Pch's user avatar
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Almost purity for perfectoid fields applied on a computation of Galois cohomology.

Let $K=\mathbf{Q}_p(\zeta_{p^{\infty}})$ and $C:=\widehat{\overline{\mathbf{Q}_P}}$. When I wrote an answer for a question a question answer, I have in my mind that "$\widehat{K}$ is a ...
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On the definition of tilting of a perfectoid field

Given a perfectoid field $K$, for example $\widehat{\mathbf{Q}_p(\zeta_{p^{\infty}})}, \widehat{\mathbf{Q}_p(p^{1/p^{\infty}})}$ or $\widehat{\mathbf{Q}_p(\zeta_{p^{\infty}}, p^{1/p^{\infty}})}$. ...
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Conductor of determinant of a Galois representation divides the conductor of the representation

I already posted this question in MO but I'm going to repost it here as well. I am studying Serre's paper "Sur les représentations modulaires de degré 2 de $\mathrm{Gal(\overline{\mathbb Q}/\...
Marta Sánchez Pavón's user avatar
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Absolute Galois group of the p-adic completion of a valuation field

Let $K$ over $\mathbf{Q}_p$ be an algebraic extension, not complete for $|\cdot|_K$ (the unique extension of $p$-adic norm $|\cdot|_p$ to $K$). (For example, $K=\cup_n\mathbf{Q}_p(\zeta_{p^n})$ the ...
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Twists and conjugates of automorphic representations

I am trying to understand the definition of RAECSDC (regular, algebraic, essentially conjugate self-dual, cuspidal), which appears in many papers on automorphy lifting. Let $\pi$ be an automorphic ...
user14411's user avatar
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Is it possible to construct geometrically the ($\phi$, $\Gamma$)-module corresponding to a $p$-adic representation coming from geometry?

The $p$-adic étale cohomology of algebraic varieties over $p$-adic fields is a fundamental subject in the study of $p$-adic representations. Moreover, thanks to the comparison theorems in $p$-adic ...
Hiroyuki Sunata's user avatar
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How to view one characters on absolute galois of a number field $F$ as an ideal group character?

Suppose I start with a number field $F$ and a character $$ \chi :G_F =\operatorname{Gal}(\overline F/F)\rightarrow {\overline {\mathbf Q}}^\times. $$ How does one get a character on ideals of $F$? ...
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Lifting of residual representation

Let $G$ be a finite group and $\bar\rho:G\rightarrow GL_2(k)$ be a residual representation, where $k$ is a field of characteristic $p$ that is the residue field of a complete discrete valuation ring $...
KS M's user avatar
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Definition unramified local representation

We can define unramified representation of $GL(2,\mathbb Q_p$) as a representation $(\pi , V) $ such that $V^K\neq \{0\}$ for $K$ the maximal compact subgroup i.e. $GL(2,\mathbb Z_p)$. However, I do ...
mathemather's user avatar
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The equivariant BSD conjecture (and the $\rho$-isotypical component)

I am trying to understand the statement of the equivariant BSD conjecture. Let $E/\mathbb{Q}$ be an elliptic curve. Let $\rho$ be a finite-dimensional irreducible Artin representation, and let $K/\...
Math-Alt's user avatar
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Universal lifting ring of Taylor-Wiles lifting

Let $\ell,p$ be two distinct primes. Let $K$ be a finite extension of $\mathbb Q_{\ell}$ with residue field $k$ of size $\# k\equiv 1\pmod p$. Let $\mathcal O$ be the ring of integers of a finite ...
user14411's user avatar
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Representations with traces in a finite field

Let $F$ be a finite field. Let $\rho:G\to GL_2(\overline F)$ be a continuous representation from a profinite group $G$, where $\overline F$ is an algebraic closure of $F$. Suppose that for all $g\in G$...
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Fixed part of the induced representation

Suppose the topological groups $h, G$ satisfy that $H < G$ and that $[G \colon H] < \infty$. Let $V$ be a $K$-vector space on which $H$ acts continuously. Then we consider the induced $G$-...
Pierre MATSUMI's user avatar
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Direct Image and the induced representation.

Suppose $f \colon \mathrm{Spec}\, L \to \mathrm{Spec}\, K$ be a finite covering. Given a smooth sheaf ${\cal F}_{\rho}$ on $\mathrm{Spec}\,L$ corresponding to the representation $\rho \colon {\pi}_1(\...
Pierre MATSUMI's user avatar
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Is a Drinfeld module over an inseparable extension isogenous to one over a separable extension?

I am reading the paper of Pink and Rutsche about the adelic openness of the image of Galois representations associated to Drinfeld modules, i.e., Pink, Rutsche, Adelic openness for Drinfeld modules in ...
Leo D's user avatar
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Absolute inertia subgroups of local fields

If $K$ is a local field, then we will denote $G_K$ the absolute Galois group of $K$, $I_K$ the inertia subgroup of $G_K$ and $P_K$ the wild inertia subgroup of $G_K$, respectively. Now, let $L/K$ be a ...
trivialquestions's user avatar
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Restriction map in semilocal Galois cohomology

Let $F/K$ be a finite extension of number fields, let $v$ be a prime of $K$ that is totally split in $F$. Call $w_1,\dots, w_n$ the primes of $F$ above $v$. Let $T$ be a $Gal(\bar{K}/K)$-module. Then ...
Fraz's user avatar
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A little proposition for Galois representations on a local field of Diamond-Shurman's book

$\def\gl{\mathrm{GL}} \def\Q{\mathbb{Q}} \DeclareMathOperator{\gal}{Gal} $ I am reading the Galois representation chapter of Diamond-Shurman "A first course in modular forms". Let $L$ be a ...
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Prove $Aut(M)$ is isomorphic to $V_4$

The full question is as follows: Take $M=\mathbb{Q}(i,\sqrt{2})$ and $\alpha=1+i+\sqrt{2}$. Prove that $G=Aut(M)$ is isomorphic to $V_4$, where $V_4$ is the group of order 4 not cyclic (Klein group). ...
cut's user avatar
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1 answer
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Is the image of Galois representaiton a Lie group over a local field of positive characteristic

Let $G={\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be the absolute Galois group over $\mathbb{Q}$ and let $\mathbb{Q}_{p}$ be the $p$-adic field. It is well-known that the image of a continuous group ...
Leo D's user avatar
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Ribet's Level Lowering Theorwm

I am reading about the modular method and I have a question about Ribet's Level Lowering Theorem. In its simplest form, it says that if an elliptic curve $E$ of conductor $N$ has no $p$-isogenies, ...
did's user avatar
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Reference request: Local Euler characteristic

I'm learning p-adic Galois representation, here is the background: Let $K$ be a p-adic field, and $G_K$ is the absolute Galois group of $K$. Let $V$ be a finite-dimensional p-adic representation of $...
Richard's user avatar
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Absolute Galois groups and representations

I've thinking about the possibility how is the image of a two dimensional complex representations of absolute galois group of $\mathbb Q$, $G_\mathbb Q$. I thought I had a proof about what the image ...
Luis Antonio Sanchez's user avatar
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Hodge-Tate decomposition for tori

I am reading the article "On the Order of the Reduction of a Point on an Abelian Variety", R.Pink (here is a link to the article). I am stuck on the proof of Proposition 1.4. Here, the ...
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Calculating Hodge-Tate numbers for the cyclotomic character

Let $\mathbb{Z}_p(1) :=T_p(\overline{\mathbb{Q}}_p^\times) =\mathbb{Z}_pt$ be the Tate module of $\overline{\mathbb{Q}}_p^\times$, i.e. the Galois group $G_{\mathbb{Q}_p}= \operatorname{Gal}(\overline{...
klein4's user avatar
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Representability of 2 dimensional $p$-adic Galois representations

Let $G = G_{\mathbb{Q}_p}$ be the absolute Galois group of $\mathbb{Q}_p$, and let $\overline{D}$ be a residual pseudorepresentation of $\mathbb{Z}_p[G]$ over $\mathbb{F}_p$. Denote by $\text{Rep}^d_{\...
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Potential good reduction and characteristic polynomial of Frobenius

Let $R$ be a discrete valuation ring with fraction field $K$ (valuation being $v$) and finite residue field $k$, $l$ be a rational prime invertible in $R$, $D(\bar{v})$ be a decomposition subgroup of ...
Doug's user avatar
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Is inertia group always commutative?

Let $K$ be a finite extension of $\mathbb Q_p$ for some prime $p$. $I_K$ is the inertia group of $\bar K/K$. Can we prove the inertia group is commutative? Actually the case is: For some ...
Richard's user avatar
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Why the semisimplicity of representation is equivalent to the semisimplicity of the element?

Consider $1\in \mathbb Z \subset \widehat {\mathbb Z}$(it's the topological generator of $\widehat {\mathbb Z}$). Let $l$ be a prime. Suppose there is a continuous representation $\rho:\widehat {\...
Richard's user avatar
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How do we explicitly compute the Galois action on etale cohomology?

The general theorems about etale cohomology are usually enough to let us compute a given $\mathrm{H}^i(X,\mathbf{Q}_\ell)$ as a $\mathbf{Q}_\ell$-vector space without too much difficulty. I would like ...
Bun's user avatar
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Converse to Proposition 2.23 in Darmon, Diamond, Taylor's FLT Notes

Can someone either prove or link me to a reference for Remark 2.24 (page 64) here? I am told that SGA7 covers this for general abelian varieties. I am wondering if a) anyone can pinpoint where in SGA7 ...
Johnny Apple's user avatar
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Method of associating Galois representation to normalized Hecke eigenform of weight 2 not work for other weights. Why?

I have read some parts of "A first course in Modular forms" to understand the process of associating a Galois representation to modular forms. In the book it is done only for weight 2 but I ...
Galois's user avatar
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Is $H^1_{et}( \overline{C},\mathbb Q_l)$ irreducible as a Galois representation?

Let $C$ be a smooth, projective, geometrically connected curve over $\mathbb Q$. We know that for any prime $\ell$ outside a finite set of primes, we have a $G_{\mathbb Q}$-representation $H^1_{et}( \...
Arkady's user avatar
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1 answer
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Why do we restrict ourselves to continuous representations of galois groups?

When studying Galois representations, we always assume that our representations are continuous. I'm new to studying these objects and am a bit struck by this assumption. What is the reasoning behind ...
Thigh High Crocs's user avatar
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113 views

Galois representation of an elliptic curve over a function field of char p.

Let $E/K$ be a non-isotrivial elliptic curve over a function field of characteristic $p$. Let $$\rho_{E,l}:Gal(K^{sep}/K)\to GL_2(\mathbf{Z}_l)$$ For $l\neq p$, the image of $\rho_{E,l}$ is a $l$-adic ...
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Understanding the action that comes with $H^1_{cont}(G_K,GL_n(\mathbb{C}_p))$?

I need to look through Sen's "Continuous Cohomology and p-Adic Galois Representations" 1990 paper, but I have confused myself. What is the $G_K$-action on $GL_n(\mathbb{C}_p)$, that makes it ...
djones's user avatar
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1 answer
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Galois group action via character

Let $G_K$ be a absolute Galois group of field $K$. Let $f$ be character of $G_K$, then, what does '$G_K$ act via $f$' mean ? (I searched the definition, but I couldn't find exactly fits this context). ...
Pont's user avatar
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Galois group of field of definition which is generated by $N$-torsion points of elliptic curve

We consider two variables $t$, $u$, and the elliptic curve $E:y^{2}=x^{3}+tx+u$ which is defined over the function field $\mathcal{K}=\mathbb{C}(t,u)$. For integer $N>1$, we define the field of ...
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Background behind Eichler's discovery of the relationship of a modular form with an elliptic curve

In Fraenkel's Love and Math (and Richard Taylor's Modular Arithmetic IAS Post https://www.ias.edu/ideas/2012/taylor-modular-arithmetic), specifically in Chapter 8 Magic Numbers, page 88., Fraenkel ...
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