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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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 ...
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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 ...
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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 ...
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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}( \...
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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 ...
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Galois cohomology and Galois representation- Related areas

Are there any interesting results we can get by combining the field of Galois cohomology and the field of Galois representations? Well, these two fields are sort of mathematical languages deals with ...
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Intuition for the generators of universal deformation rings

While I'm studying about modularity lifting theorems, I faced two theorems by Ramakrishna about the structure of universal deformation rings. Theorem 1 Let $p > 2$, $K/\mathbb{Q}_{p}$ a finite ...
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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 ...
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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). ...
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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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Discovery of the relationship of the Theorem 18.5 in Ash & Gross's "Fearless Symmetry" book

In Ash & Gross book: Fearless Symmetry, Chapter 18, the following Theorem is stated: THEOREM 18.5: Let q be a prime other than p that is unramified for the Galois representation ψ. (This will be ...
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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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Representations with finite character in Langlands

I'm trying to understand some basic Langlands's, and whilst reading this by E.Frenkel, he says (at the bottom of page 5) that (in the function field case), it suffices to show that isomorphism classes ...
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Why does having an integral model make étale cohomology unramified?

Let $K$ be a number field, and $\mathcal O$ its ring of integers. Fix a finite set $S$ of rational primes, and let $\mathcal O_S$ be the ring of integers with these primes inverted. Let $X$ be a ...
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Do ramification groups contain non-central abelian normal subgroups?

I am studying the proof of integrality of the conductors of Galois representations from these notes, and I have hit a roadblock in a step of the proof of Proposition 3.1.40 (page 57). The setting is ...
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Proposition 10.4 in Neukirch's Algebraic Number Theory

I am stuck on a small detail in the proof of Proposition 10.4(iii) from Chapter VII of Neukirch's Algebraic Number Theory. For a Galois extension of number fields $L|K$ and a representation $(\rho, V)$...
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Another question concerning Lemma 1.12 of Wiles's proof of Fermat's Last Theorem

This is a follow-up on my previous question. So, at the risk of repeating myself, let me give some notation: Let $k$ be a finite field of characteristic $p\neq 2$ (in fact, one only needs to consider ...
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Confusion concerning Lemma 1.12 in Wiles's proof of Fermat's Last Theorem

Let $k$ be a finite field of characteristic $p\neq 2$ (in fact, one only needs to consider the case $p\in\{3,5\}$), let $\Sigma$ be a finite set of primes containing $\infty$ and $p$, and $$\rho_{0}:{\...
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Artin $L$-function is well defined

Here is what I got so far. Let $\rho: Gal(L/K)\rightarrow GL(V)$ be an Artin representation. Let $\frak{p}$ be a prime of $K$, $\frak{P}$ a prime of $L$ lying above $\frak{p}$ and denote by $D_\frak{...
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On ${\Bbb Z}/p{\Bbb Z}$-torsor.

I exemplify the previous question On ${\Bbb Z}/m{\Bbb Z}$-torsors. Let $k$ be a field of charactersitic $p$ and suppose that one has a character $\chi \colon {\mathrm{Gal}}(\overline{k}/k) \to {\Bbb Z}...
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Meromorphic continuation of L-functions

I am following these notes and on page 2 the claim is that if we have an $L$-function $$L(s) = \sum_{n=1}^{\infty}\frac{a_n}{n^s}$$ with $a_n=O(n^r)$ and if $L$ has a meromorphic continuation and ...
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Pure Galois representation

I have two questions about pure Galois representations. Let $K$ be a number field and consider a continuous $\ell$-adic Galois representation $\rho$. Assume that we have a finite Galois representation ...
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Left versus Right regular representations.

Let $G$ be a finite group. $G$ can bear the so-called regular representation. Let $\chi_g(h) \colon= \delta_{g,h} ~ {\mathrm{for}} ~ h \not= g$. Let $X \colon= {\mathrm{the\, vector\,space\,of\,...
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Shimura reciprocity law

Let $X_s$ be the compact modular curve of level $\Gamma_0(N)\cap\Gamma_1(p^s)$, with $N\in\mathbb{N}$ and $p$ prime, $(N,p)=1$. Then noncuspidal points on $X_s$ correspond to triples $(E,\frak{n},\pi)$...
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Textbook of p-adic hodge theory

It seems that p-adic hodge theory is essential ingredient in arithmetic geometry. I would like to learn p-adic hodge theory, so I have searched for textbooks of p-adic hodge theory, but I wasn't able ...
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Properties of the Galois Representation Attached to a Weight 2 Cusp Form

The paper here posits on page 86 that it is Shimura who proved in "Introduction to the Arithmetic Theory of Automorphic Functions" that for a prime $p$, the $p$-adic Galois representation ...
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Necessary and Sufficient Conditions for the mod-$l$ Representation Attached to an Elliptic Curve to be Unramified/Flat

I am seeking References for facts from chapter I of Cornell-Silverman-Stevens, "Modular Forms and Fermat's Last Theorem." On page 6 of the text, the authors provide in Theorem 2.11 some &...
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Local systems on a punctured line

How to describe the category of local systems on $\mathbb{G}_{m, \mathbb{Z}}$ in different topologies? I think that in etale topology there is a version of Riemann-Hilbert correspondence which says ...
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computing Frobenius in $Gal(\mathbb{Q}(i, \sqrt[4]{2})/\mathbb{Q}) \simeq D_8$ example in Weinstein survey article

The following is taken from this survey article Weinstein Reciprocity Laws by Jared Weinstein. My question is how do I derive the result boxed in green - the conjugacy class of $Frob_p$ in the Galois ...
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Show the stabilizer is an open subgroup [closed]

Let $V$ be a $l-$adic representation of a Galois group $G$ where $V$ is equipped with the $l-$adic topology. Let $T_0$ be a lattice in $V$ and $H=\{g \in G \mid g(T_0)=T_0\}$. Show $H$ is an open ...
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De Shalit's lemma in R = T.

In Wiles' celebrated paper where any semi-stable elliptic curve $E$ over ${\Bbb Q}$ is modular, Theorem $0.3.$ therein assumes that either $E$ is good or multiplicative reduction at $3$. This ...
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Image of projective representations of Weil group

Let $F$ be a number field and let $\sigma : G_{F} = \mathrm{Gal}(\overline{F}/F) \to \mathrm{GL}_{2}(\mathbb{C})$ be a continuous representation of Galois group. Since $G_F$ is compact, the image of $\...
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2 votes
1 answer
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Image of Galois representation of ordinary elliptic curve

Let $K$ a local field, $E/K$ elliptic curve, $k$ the residue field, $p=\operatorname{char}(k)$. Assume $\tilde{E}/k$ is a good reduction and assume the elliptic curve has nonzero Hasse invariant, i.e ...
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What is $\text{Frob}\in \text{Gal}(\overline{K}/K)$?

Let $\overline{\rho}: \text{Gal}(\overline{K}/K)\to \text{GL}_2(\mathbb{F}_q)$ be an unramified Galois representation at a place $v$ of $K$. Since $\ker(\overline{\rho})$ is closed, it corresponds to ...
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Action of decomposition subgroup on roots of unity

Let $G_q$ be the decompostion subgroup $\text{Gal}(\overline{\mathbb{Q}}_q/\mathbb{Q}_q)$ where $q$ be a prime. Let $p$ be a prime and $p \ne q$ and we take $\mu_p$ be the group $p$-th roots of unity ...
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$\dim(H^1(G_K,\mathbb{Q}_p(1)))=1+[K:\mathbb{Q}_p]$ when $\ell=p$ and $1$ otherwise? [closed]

Let $K$ be a finite extension of $\Bbb{Q}_{\ell}$ and let $G_K=\mathrm{Gal}(\overline{K}/K)$. Then is the following true, by the Euler-Poincare characteristic formula? $\dim(H^1(G_K,\mathbb{Q}_p(1)))=...
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Canonical action of ${\mathrm{Gal}}(K/{\Bbb Q})$.

Let $K/{\Bbb Q}$ be a finite Galois extension, and ${\mathrm{Gal}}(K/{\Bbb Q})$ the galois group. The ${\mathrm{Gal}}(K/{\Bbb Q})$ module ${\text{H}}^1({\mathrm{Gal}}(\overline{\Bbb Q}/K), {\Bbb Z}/n{\...
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1 answer
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Weight of dual Galois representation

Let $K$ be a number field. We say that an $\ell$-adic Galois representation is of pure weight $n$ if for almost all primes $\mathfrak{p}$ the eigenvalues of $\text{Frob}_{\mathfrak{p}}$ are of ...
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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 $\Phi(GSpin_{2n + 1})$ ...
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Local polynomial of Galois representation restricted to subextension

Let $K$ be an extension of $\mathbb{Q}_p$. Consider an Galois representation $\rho: G_K \to GL_2(\mathbb{C})$ and for any finite extension $M/K$ we call $$P(\rho|_M,T) = \det(1-\operatorname{Frob}_{M}^...
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What is the algebraic group whose $\mathbb Q_p$-rational points are the group $\mathrm{GU}_n(\mathbb Q_p)$?

For any field $K$, I denote by $\Gamma_K = \mathrm{Gal}(\overline K/K)$ its absolute Galois group. Let $p$ be a prime number and let $\mathbb Q_{p^2}$ denote a quadratic unramified extension of the ...
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A question on Robba ring

With notations as in the question: some questions about the Robba ring. Moreover, we define a new operator over the Robba ring as follows. Put $$c=\frac{pE(u)}{E(0)}$$ and define a Frobenius map $\...
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Commutative algebra details on patching when proving $R = \mathbb{T}$ theorem (Calegari-Geraghty Paper)

I've been working on understanding the proof of Fermat's last theorem and now focusing on the patching technique for modularity lifting. I found that the patching technique described in the paper ...
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5 votes
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On ${\Bbb Z}/m{\Bbb Z}$-torsors.

I would like to know the explicit construction of ${\Bbb Z}/m{\Bbb Z}$-torsor $Y$'s over a scheme $X$. It is explained that $X$ are classified by $H_{et}^1(X, {\Bbb Z}/m{\Bbb Z})$, which is far from ...
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7 votes
1 answer
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some questions about the Robba ring

Notations and definitions Let $p$ be a prime integer, $k$ be a perfect field of characteristic $p$ and $W(k)$ its ring of Witt vectors. Definition 1 We put $$ \mathcal{R}_r=\bigg\{ \sum_{i\in \...
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How to define the Stiefel-Whitney class of a complex orthogonal representation?

Background: One of the main objects of interest in the theory of $L$-functions is the root number, a complex number of modulus one which appears in the functional equation. In general, a root number ...
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Field cut out by a character

Let $p$ be an odd prime and let $\Delta:=\textrm{Gal}(\mathbb Q(\zeta_p)/\mathbb Q)$, where $\zeta_p$ is a primitive $p$-th root of unity. Consider the character: $ \chi: \textrm{Gal}(\overline{\...
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3 votes
1 answer
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Morphism of admissible $1$-dimensional $(\varphi,N)$-module

In these notes in section 7.3 admissible one-dimensional $(\varphi,N)$-modules are classified. Let $K$ be a $p$-adic field. Then for any $\lambda\in K_0$, they give an element of $MF_K^{ad}(\varphi,N)...
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4 votes
1 answer
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On understanding $\mathrm{ad}^0 \overline{\rho}$ and $\mathrm{ad}^0 \overline{\rho}(1)$ in Taylor-Wiles method

I'm currently learning Taylor-Wiles method and modularity lifting and comming up with following difficulties, which I think is based on understanding how (global and local) Galois groups act on $\...
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