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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45 views

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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1answer
32 views

Existence of small extensions

Let $\mathsf{CNL}$ be the category of complete noetherian local rings with (fixed) residue field $\mathbb{F}$. The morphisms are local homomorphisms fixing the residue field. The full subcategory of ...
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30 views

Does representability of the subfunctor coherent with taking quotients?

I'm currently learning the deformation of Galois representations by reading Gouvêa's note. Let $\mathsf{CNL}$ be the category of complete noetherian local rings with (fixed) residue field $\mathbb{F}$....
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96 views

Fixed field of $G(K/\Bbb Q)\cap V$

I've been trying to solve the following question, but I wasn't able to. Given $p\in\Bbb Q[x]$ an irreducible quartic polynomial, and $a,b,c,d$ its roots, consider $K=\Bbb Q(a,b,c,d)$ its splitting ...
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An example of $p$-divisible group

In page 76 of Demazure's book "Lectures on $p$-divisible groups". The formal group $G^{\lambda}$ defined by the exact sequence $$ 0\to G^{\lambda} \to W(p) \xrightarrow{F^r-V^s} W(p) $$ is ...
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Statement about Weil representations being unramified over a certain extension

Let $K = \mathbb{Q}_p$ and $\rho: W_K \to \operatorname{GL}(V)$ be a Weil representation over a finite-dimensional complex vector space $V$ (where $W_K$ denotes the Weil group over $K$). Let $L/K$ be ...
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1answer
31 views

Image of continuous Galois representation constructed by Deligne corresponding to a eigenform.

It is a well known theorem that given a eigneform $f$ of weight $k$ and level $N$, Deligne, Serre, Shimura and others have constructed continuous $\ell$-adic Galois representation $\rho_f : \mathrm{...
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38 views

Galois representation from the “group algebra” point of view

Let $G$ be a group (NOT necessarily finite), $F$ be a field, then we know that the category of all $F$-representations of $G$, denoted by $(\mathsf{Rep}_{F}(G))$, is equivalent to the category of $FG$-...
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34 views

What about the algebraic structure of $T_p(G) \otimes_{\mathbb{Z}_p}A$?

Let $G$ be a $p$-divisible group over the ring of $p$-adic integers $O_K$ of $p$-adic field $K$. The $p$-adic Tate module $T_p(G)$ of $G$ is rank $1$ free $\mathbb{Z}_p$-module. Then $T_p(G) \otimes_{...
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1answer
89 views

On splitting of the tensor product.

Let ${\Bbb C}$ be a complex number field. We shall define the two-variable polynomial ring $R$ as follows$\colon$ \begin{equation*} R \colon = {\Bbb C}[X,Y] = {\Bbb C}[X] \otimes_{\Bbb C}{\Bbb C}[Y] \...
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72 views

Brauer–Nesbitt theorem — Does Finite-dimensionality matter?

The Brauer–Nesbitt theorem that many literatures cite is stated as follows: (BS.1): Let $KG$ be a group algebra of a finite group $G$ over a field $K$. Let $M,N$ be $KG$-modules. Suppose [A condition ...
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71 views

Shimura curves over totally real number fields

Suppose that $F$ is a totally real number field such that $[F:{\Bbb Q}]$ is odd. Then we shall choose the quaternion algebra $D$ everywhere unramified at finite places and at all but one infinite ...
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1answer
46 views

p-adic cyclotomic character on Frobenius at p

Let $p$ be a prime. From what I understand, the p-adic cyclotomic character $\varepsilon:Gal(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathbb{Z}_p^\times$ is given by choosing a compatible sequence of $p$...
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Uniqueness of the Eichler order

Let ${\mathrm{M}}_2(\widehat{{\cal O}_K})$ be the $2 \times 2$ matrices over the finite adele of the full integer ring ${\cal O}_K$ of a totally real #-field $K$. For the quaternion algebra $D_K$ over ...
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An example of $p$-divisible group from the book of Demazure.

I have some questions when I read the book "Lectures on $p$-divisible groups". Precisely, my questions are in (page 76) chapter IV, section 3: The $F$-spaces $E^{\lambda}, \lambda\geq 0$. ...
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1answer
39 views

Characters of the absolute Galois group and their restrictions

I am reading this paper Cremona et. al. In section 3 (pages 5-6), they talk about characters more specifically about Dirichlet characters and characters of the absolute Galois group, $G_{\mathbb{Q}}$. ...
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37 views

Why is the following map $\mu : G_K \to M_d(\Bbb C_K) : g \mapsto a_{ij}(g)$, where $g(w_j) = \sum_{i} a_{ij}(g)w_i$, continuous?

Let $K$ be a p-adic field and let $\Bbb C_K$ be the completion of its algebraic closure. Let $G_K := \operatorname{Gal}(\overline{K}/K)$. Fixing a basis $\lbrace w_1, \dots, w_d\rbrace$ of a $\Bbb C_K$...
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Skew-symmetricity of the Cup product

Let $f, g \in H^1(G, {\Bbb Z}/n{\Bbb Z})$ be two elements. Consider the cup-product \begin{equation} ∪ \colon H^1(G, {\Bbb Z}/n{\Bbb Z}) \times H^{1}(G, {\Bbb Z}/n{\Bbb Z}) \to H^2(G, {\Bbb Z}/n{\...
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1answer
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Matrix calculation used to determine local deformation ring

Let $C$ be a local ring with maximal ideal $m_C$ and residue field isomorphic to a finite extension of $\mathbb{F}_l$. Suppose $q$ is a power of a prime $p \neq l$ and suppose $A,B,P,Q,R,S \in m_C$ ...
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1answer
103 views

Finite Galois representations are Geometric?

A famous conjecture, the Fontaine-Mazur conjecture, predicts which $p$-adic Galois representations of a number field "come from geometry" are a subquotient of the (Weil) cohomology of a ...
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1answer
83 views

Galois representation being unramified is Galois local

Let $K$ be number field and $\rho:G_K\rightarrow \text{Gl}(V)$ a Galois representation. Let $\nu$ be a place of $K$ (non-archimedean if it helps/is necessary). We say that $\rho$ is unramified at $\nu$...
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About a specific galois extension inside $C^{\flat}$, related with Fontaine's period rings.

Let $K$ be a complete discrete valuation field of characteristic $(0, p)$ with perfect residue field $k$. Fix $\overline{K}$ an algebraic closure of $K$ and with absolute galois group $G_K$. Denote $\...
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1answer
71 views

$\ell$-adic Representations of Finite fields

I am recently read the book "Theory of $p$-adic Galois Representations" written by Fontaine and Ouyang. (Here is the link to this book: http://staff.ustc.edu.cn/~yiouyang/galoisrep.pdf) Let $...
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Literature request: Galois representations

Not really much to say here; I'm looking for an "introductory text" on the theory of Galois representations. I'll be attending a seminar on the local Langlands correspondence for $\text{GL}...
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82 views

Bloch-Kato-Selmer group of one-dimensional representation.

Let $L/\mathbb{Q}$ be a finite extension and let $V$ be a one-dimensional $L$-linear representation of $G_{\mathbb{Q}}$ which is given by $\chi\rho^*\kappa^n_{cyc}:G_{\mathbb{Q}}\rightarrow L^\times$, ...
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86 views

Local Galois group action on étale cohomology

Consider a scheme $X$ over $\mathbb{Q}$. By the base change theorem for the étale cohomology, we have $$H^i_{ét}(X_{\bar{\mathbb{Q}}},\mathbb{Z}/\ell^n)\cong H^i_{ét}(X_{\mathbb{C}_p},\mathbb{Z}/\ell^...
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57 views

How to show two Galois groups $G_1$ and $G_2$ are isomorphic to each other?

This is a question about group representation. This is not my close area but I need to understand the following. Let $E_1$ and $E_2$ be two elliptic curve over $\mathbb{Q}$. Let $\Lambda_1$ and $\...
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27 views

Duality for modules over a ring $A$.

Let $R$ be a complete regular local ring and $A_n$ be a ring defined by $A_n \colon= R/{\frak m}_R^n$. Suppose that $A_n$-module $M_n$ is defined as a free module $M_n \colon= A_ne_1 \oplus \cdots \...
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1answer
138 views

Elliptic curves with supersingular reduction have irreducible mod $p$ representations?

Let $E$ be an elliptic curve over $\mathbb Q_p$ and suppose that $E$ has good reduction at a prime $p$. I read here that if $E$ has ordinary (resp. supersingular) reduction at $p$ then the mod $p$ ...
5
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1answer
48 views

Fixed subfield of symmetric rational functions $K(s_1,\ldots,s_n)$ under $A_n$

Let $K$ be a field of characteristic $\operatorname{char} K\neq 2$, and let $L=K(x_1,\ldots,x_n)$ be the field of rational functions of $n$ variables with coefficients from $K$. Denote $F=L^{S_n}=K(...
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1answer
73 views

On prime ideal and irreducible ideal in R[X].

For a noetherian domain $R$, an irreducible ideal $I$ implies $\sqrt{I}$ is a prime ideal. Irreducible implies primary, but not always vice versa. That said, I would like to ask whether the following ...
2
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1answer
140 views

An isomorphism related to Fontaine's periods rings

Let $\Lambda$ be a ring, $V$ a $\Lambda$-algebra complete Hausdorff for the $p$-adic topology where $p$ a a prime integer. Assume that $V/pV$ is semiperfect (i.e., the Frobenius map is surjective). ...
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55 views

Why do the Galois and Hecke action commute on the Picard group of the modular curve $X_1(N)$?

In chapter 9 of Diamond-Shurman's book A First Course in Modular Forms, when they construct the Tate module associated to the modular curve $X_1(N)$, they state that the Galois action and Hecke action ...
2
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1answer
41 views

Definition of non split Cartan subgroup

What is the definition of the non split Cartan subgroups of $GL_2(\mathbb{F}_p)$? And what are the explicit expression of a matrix of this subgroups? I read on "Modular Functions of One Variable III" ...
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1answer
48 views

About Galois representation, which Galois extension should i use?

In the book "Rational Points on Elliptic Curves by J.Silverman and J.Tate" it is defined a representation $$ \rho_n:Gal(\mathbb{Q}(E[n])/\mathbb{Q})\longrightarrow GL_2(\mathbb{Z}/n\mathbb{Z})\hspace{...
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Is the Galois condition necessary in order to define an action on an elliptic curve?

At page 186 of the book "Joseph H. Silverman, John Tate - Rational points on elliptic curves-Springer-Verlag (1992)" there is a proposition where for any Galois extension $K/\mathbb{Q}$ it is defined ...
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Reference request for equivalence of reducibility of $\ell$-adic Galois representation and isogeny of degree $\ell$ on elliptic curve

Let $E$ be an elliptic curve over $\mathbb{Q}$. For each prime $\ell$, the action of $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on $E[\ell]$ (the group of $\ell$-division points of $E$) defines a ...
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1answer
74 views

$p^n$ power torsion and $p^{n+1}$ power torsion defined over the same field

Let $E$ be an elliptic curve over $\mathbb{Q}$ with no rational $p$-torsion, and with good reduction over $\mathbb{Q}$. (I'm not sure if the good reduction is really necessary here.) For each $n$, we ...
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If $\rho$ is a mod $p$ irreducible representation of $G_{\mathbb{Q}_p}$, why is it tamely ramified?

Fix a prime $p$ and let $\rho:G_{\mathbb{Q}_p}\rightarrow GL_2(\overline {\mathbb{F}}_p)$ be an arbitrary continuous representation. I found the following statement in a paper on non-ordinary modular ...
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1answer
71 views

Why $\mathbb{Q}(x_1,y_1,…,x_n,y_n)/\mathbb{Q}$ is a Galois extension? where $E[m]=\{(x_1,y_1),…,(x_n,y_n)\}$ is the m-torsion group.

Let be E an elliptic curve and $E[m]=\{(x_1,y_1),...,(x_n,y_n)\}$ the m-torsion group. Let be $K=\mathbb{Q}(x_1,y_1,...,x_n,y_n)$, why $K/\mathbb{Q}$ is a Galois extension?? I see one proof in the ...
2
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1answer
56 views

Details in Deligne-Serre “Formes modulaires de poids 1”

There are specific details which I'm a little stuck on in Deligne and Serre's paper on attaching Galois representations to modular forms of weight 1. In the proof of Lemma 8.3, they use the ...
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1answer
88 views

Decomposition group of $G_{K,S}$

Let $K$ be a number field and fix a prime $p$. Let S be a finite set of places of $K$ containing the places above $p$ and the infinite places. Let $G_{K,S}$ be the absolute Galois group of the maximal ...
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41 views

Kronecker-Weber à la Wiles

I've been studying the notes of E.Kowalksi on the Course by Tunnell on Wiles's proof of Taniyama-Shimura. As a baby case, they reprove the Kronecker-Weber Theorem. However, I'm confused by the last ...
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1answer
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A question on finding an inverse image of $\varphi-1$ over some $(\varphi, \Gamma)$-module, from an article by Cherbonnier and Colmez.

I am reading the article THEORIE D’IWASAWA DES REPRESENTATIONS p-ADIQUES D’UN CORPS LOCAL by Cherbonnier and Colmez https://webusers.imj-prg.fr/~pierre.colmez/CCjams.pdf Question: I have a question ...
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1answer
53 views

Which values determine the whole Galois representation of an elliptic curve over the p-adics with potentially good reduction?

Let $K$ be an extension of $\mathbb{Q}_p$ and $E$ be an elliptic curve over $K$ with potentially good reduction, i.e. there exists a finite extension $L/K$ such that $E/L$ has good reduction. Let $\...
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1answer
43 views

(In)dependence of the conductor of a Galois representation and the choice of l

In the work of A. P. Ogg, Elliptic Curves and Wild Ramification, he proves that the conductor of an elliptic curve is independent of the choice of $\ell$. That is, for example, if $E$ is an elliptic ...
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25 views

Allowed ramification of deformations of Galois representations

I am trying to learn about deformations of degree 2 Galois representations mod $p$ and get a grasp of simple intuitions on examples. In basic references the explicit examples of universal deformation ...
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54 views

Galois Representations and Eigenvalues

Suppose you have two ($\ell$-adic) Galois representations $V$ and $W$ and you know that each Frobenius-eigenvalue (at unramified primes) for $V$ is also one for $W$. Is this enough to obtain a Galois-...
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32 views

How to represent a Galois Field with polynomials

I'm given the task of representing a galois field, specifically $GF(107^2)$, by polynomials modulo a irreducible polynomial (which I'm given) in $Z_{107}[x]$. I'm confused on how I'm supposed to go ...
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81 views

Compute the level of the Galois representation

In the computation of the conductor of an elliptic curve usually we use the Tate algorithm to determine the singular fibre of an elliptic curve, but what should we do when we have a variable in the ...

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