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5
votes
1answer
98 views

Artin conductor of a character and factorisation through $(\mathbb{Z} / N\mathbb{Z})^{*}$

This is from Serre's paper on modular representations of degree $2$ of $Gal(\bar{\mathbb{Q}} : \mathbb{Q} ) $. We consider a representation $\rho : Gal(\bar{\mathbb{Q}}:\mathbb{Q}) \rightarrow ...
1
vote
1answer
55 views

Motivation for the definition of the Artin Conductor of a representation

I'm trying to figure out what the Artin Conductor of a representation is using Chapter IV.$2\text-3$ of Serre's Local Fields, and I'm struggling to understand the motivation behind its definition. ...
4
votes
1answer
64 views

Galois representations and isogenies of elliptic curves

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 ...
6
votes
2answers
65 views

The mod $p$ Galois representation of the Frey curve is unramified away from $2, p$

Given a hypothetical solution to Fermat's last theorem for $p \ge 5$ $$a^p + b^p + c^p = 0$$with $a \equiv -1 \pmod 4$, $b$ even, we can write down the Frey Curve$$E: y^2 = x(x-a^p)(x+b^p)$$which has ...
3
votes
1answer
49 views

image of Kummer isomorphism

Let $K$ be a finite extension of the field of $p$-adic numbers which contains the $p$-roots of unity. There is an isomorphism $K^{\times} / (K^{\times})^p \to \mathrm{Hom}(\mathrm{Gal} (\bar{K} / K), ...
10
votes
1answer
105 views

In what sense is $p$-adic Hodge theory related to ordinary (complex) Hodge theory?

I started reading about $p$-adic Hodge theory in the notes of Brinon and Conrad. I quote (page 7): The goal of p-adic Hodge theory is to identify and study various “good” classes of $p$-adic ...
0
votes
0answers
55 views

How to construct this modular Galois representation?

I want to know how to construct this modular Galois representation: The $\bmod p$ representation is semi-simple. For any lattice $T$ , $\dfrac{T}{p^{3}T}$ does not have a $G$-stable cyclic subgroup ...
1
vote
1answer
47 views

Complex Galois Representaions

I'm trying to understand 1 and 2 dimensional complex representations, induced representations and associated Artin L-functions for a project. I'm finding it hard to find appropriate material to help ...
6
votes
2answers
150 views

Galois theory, had it solved any major problems beside its original applications to classical problems?

Galois theory, had it solved any major problems beside its original (classical) applications to roots of a fifth (or higher) degree polynomial equation (solvable algebraic equations and constructible ...
1
vote
1answer
46 views

Surjectivity of p-adic representation

Let $E/\mathbb{Q}$ be an elliptic curve. For a prime $p$, we have the mod $p$ representation \begin{equation*} \bar{\rho}_{E,p}: G_{\bar{\mathbb{Q}}/\mathbb{Q}} \rightarrow Aut(E[p]) \end{equation*} ...
1
vote
1answer
30 views

Image of the p-Frobenius endomorphism under a mod p Galois representation.

Let $E/\mathbb{Q}$ be an elliptic curve and $p$ a prime such that $E$ has ordinary redcution at p. Further, let $$\rho_{E,p}:{\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to {\rm ...
3
votes
1answer
91 views

What is : $ \ \mathrm{Gal} ( \overline{ \mathbb{Q} } / \mathbb{Q} ) $?

In a book, I find the following thing : The natural homomorphism $ \mathrm{Gal} ( \overline{\mathbb{Q}} / \mathbb{Q}) \to \displaystyle \lim_ {\longleftarrow n} \mathrm{Gal} (K_n / \mathbb{Q}) $ is ...
2
votes
1answer
54 views

Computing the trace of the image of Frobenius

Let $E/\mathbb{Q}$ be an elliptic curve, $p$ and $q$ distinct primes and $e$ a positive integer. Fixing a basis for the $p^e$ torsion, we get a natural Galois representation $$\rho_{E,p^e}:{\rm ...
8
votes
1answer
193 views

What is the intuition behind the Fontaine-Mazur Conjecture?

The Fontaine-Mazur conjecture (over $\textbf{Q}$ for simplicity) says that a (continuous irreducible) Galois representation $$ \rho: \text{Gal}(\overline{\textbf{Q}}/\textbf{Q}) \to ...
4
votes
1answer
78 views

Can one check by hand whether the Tate module of an elliptic curve is semi-simple

Let $E$ be an elliptic curve over $\mathbb Q$, and $\ell$ a prime number. Then, the $\ell$-adic Tate module $V_\ell(E)$ of $E$ is semi-simple as a $\mathbb Q_\ell$-representation of ...
6
votes
1answer
196 views

Serre's Modularity Conjecture — Weight

I was reading Serre's paper "Sur les Représentations Modulaires de Degré $2$ de Gal($\bar{\mathbb{Q}}/\mathbb{Q}$)" where he states his modularity conjecture (which is now a theorem). Following his ...
2
votes
1answer
58 views

Characterisation of Galois Group with the action of $\sigma \in S_n$ on the roots

Let $f \in K[X]$ be irreducible and separable with roots $x_1,...,x_n$ in a splitting field $L$ of $f$ over $K$. We identify $\text{Gal}(L|K)$ with $\text{Gal}(L|K)\cong G\subset S_n$. How can I see ...
0
votes
0answers
40 views

Galois representation associated to a number field

I think I'm missing something completely trivial. I want to know how to compute the Galois representation associated to an extension of $p$-adic fields. Let $p$ and $q$ be odd prime numbers. Fix ...
2
votes
1answer
49 views

invariants of a representation over a local ring from the residual representation

Let $(R, \mathfrak m)$ be a local ring (not necessarily an integral domain) and $T$ be a free $R$-module of finite rank $n\geq 2$. Let $\rho: G \to \mathrm{Aut}_{R\text{-linear}}(T)$ be a ...
3
votes
1answer
140 views

Cyclotomic Character

I have a couple of questions concerning the cyclotomic character. For the moment I know very little about the mod $\ell$ cyclotomic character, namely that ...
5
votes
2answers
149 views

Importance of continuity of Galois representations

So for a one dimensional Galois representation $\rho: G_{\Bbb Q} \to \mathbb C^{\times}$, I know that it must factor through the abelianization of $G_{\Bbb Q}$, which by the Kronecker-Weber theorem is ...
5
votes
2answers
166 views

Elements of order 2 in the absolute Galois group

So I remember reading once that the only element of $G=Gal(\overline{\Bbb Q} / \Bbb Q)$ that we understand is complex conjugation. Suppose we fix an embedding of $\overline{\Bbb Q}$ into $\Bbb C$. ...
0
votes
1answer
51 views

Tate module of linear algebraic group

Let $G$ be a (smooth, connected, geometrically integral) commutative linear algebraic group over $\mathbf F_q$. Just as for abelian varieties, we can define the $\ell$-adic Tate module $$ T_\ell G ...
3
votes
1answer
101 views

$p$-adic representation and $p$-adic analytic group ($p$-adic Lie group)

A $p$-adic representation of a group $G$ is continuous group homomorphism $$\rho: G \to GL_n(\mathbb{Q}_p)$$ How are those representations related to $p$-adic analytic groups (= $p$-adic Lie ...
3
votes
1answer
61 views

Identifying the residue field of $\overline{\mathbb{Q}}_p$ with $\overline{\mathbb{F}}_p$

Could anyone epxlain to me how to identify the residue field of $\overline{\mathbb{Q}}_p$ with $\overline{\mathbb{F}}_p$ ? I am basically trying to go from a $p$-adic galois representation of ...
1
vote
1answer
72 views

Image of Frobenius

So here is the setting: I have a Galois extension $L/K$, and a prime ideal $\mathfrak{P}$ of $\mathcal{O}_L$ lying over a prime ideal of $\mathcal{O}_K$. There is a map $ D_{\mathfrak{P}} \rightarrow ...
8
votes
1answer
171 views

Absolutely irreducible representations of the absolute Galois group of $\mathbb{Q}_p$

Let $p$ be a prime number. Denote by $G$ the absolute Galois group of (a finite extension of) $\mathbb{Q}_p$. Let $\ell$ be a prime number. For $\ell= p$, I guess it is well known that the ...
4
votes
1answer
267 views

Galois representations attached to modular forms of weight one

Why are the Galois representations attached to weight one eigenforms by Serre-Deligne complex representations? In weight $k\geq 2$, the Galois representations constructed by Eichler-Shimura and ...
0
votes
0answers
48 views

On a abelian representation of galois group

My question is quite elementary. I am wondering all irreducible abelian Galois representation of $Gal(\bar{Q}/Q)$ should be character.(i.e 1-dimensional). I think it should be ture. But since no ...
1
vote
1answer
74 views

Odd representations over absolute Galois Group of totally real field

(This might be an easy quesiton but I'm new to representations and could use a helpful pointer or two) Let $G_{\mathbb{Q}}= \text{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$ be the absolute Galois ...
6
votes
0answers
74 views

Generalization of Kummer isomorphism?

Let $p$ be a prime number and denote by $\mathbb{F}_p(1)$ the one dimensional vector space over $\mathbb{F}_p$ endowed with an action of $G:=Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p)$ via the mod $p$ ...
4
votes
0answers
84 views

Surjective map in Galois cohomology?

Let $p$ be a prime number, $\mathbb{F}_p$ the field with $p$ element and $\omega$ the mod $p$ cyclotomic character. Let $K$ be a finite extension of $\mathbb{Q}_p$ (the field of $p$-adic numbers) and ...
1
vote
1answer
2k views

multiplication in GF(256) (AES algorithm)

I'm trying to understand the AES algorithm in order to implement this (on my own) in Java code. In the algorithm all byte values will be presented as the concatenation of its individual bit values (0 ...
3
votes
1answer
143 views

bests book of representation theory for algebraic number theorists

I am looking for some of the best books on representation theory for an algebraic number theorists> I would prefer a book that is more number theoretical (e.g, galois representations, p adic ...
3
votes
0answers
46 views

computation of an $H^2$

Let $p$ be a prime number and $\mathbb{Q}_p$ the field of $p$-adic numbers. Let $G_p$ be the absolute Galois group of $\mathbb{Q}_p$ and fix an absolutely irreducible representation $$ \rho : G_p \to ...
10
votes
1answer
227 views

Galois representations and normal bases

I am not very familiar with the theory of Galois representations, but I do know a bit about both Galois theory and representation theory. Recently I learned about the notion of a normal basis for a ...
3
votes
1answer
121 views

Walsh spectrum of a function defined over Galois rings

Let $GR(p^2,m)$ be the Galois ring with $p^{2m}$ elements and characteristic $p^2$. Let $Z^m_{p^2}$ be the cross product of $m$ copies of $Z_{p^2}$ which is the set of integers from zero up to ...
4
votes
2answers
265 views

commuting algebra of an irreducible representation

Let $V$ be a finite-dimensional vector space and $\rho$ an irreducible abelian representation of $G$ on $V$. Is the centralizer of $\rho(G)$ in $End(V)$ necessarily a (commutative) field? (In ...
4
votes
0answers
117 views

Ribet's proof of open image for elliptic curves

In http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183555477, Ribet gives a proof of Serre's open image theorem for elliptic curves using ...
8
votes
0answers
114 views

Reference for l-adic Lie algebras

I don't know much at all about Lie algebras or representation theory, and I'm trying to read Ribet's `Review of Abelian l-adic Representations and Elliptic Curves'. Is there a standard reference for ...
9
votes
1answer
200 views

Connection between the $L$-function of a modular form and the $L$-function of its associated $\ell$-adic representation as defined by Bloch-Kato

Let $f\in\mathcal{S}_k(\Gamma_1(N),\epsilon)$ is a normalized eigenform for all the Hecke operators, with character $\epsilon$, $k\geq 2$, and assume the $q$-expansion of $f$ has rational ...
4
votes
1answer
86 views

Local invariants of the discrete Galois module associated to a $p$-ordinary newform

Let $f=\sum_{n=1}^\infty a_nq^n$ be a $p$-ordinary newform of weight $k\geq 2$, level $N$, and character $\chi$, and let $\rho_f:G_\mathbf{Q}\rightarrow\mathrm{GL}_2(K_f)$ be the associated $p$-adic ...
7
votes
2answers
187 views

Clarifying a comment of Serre

Let $\rho_{\ell}$ be the "mod $\ell$" Galois representation associated to an elliptic curve $E/K$ (i.e., corresponding to the action of Galois on the $\ell$-torsion points). Serre proved that in the ...
1
vote
1answer
117 views

Lifting additive characters

Let $K$ a finite extension of $\mathbb{Q}_p$ ($p$ prime different from 2) and let $G_K$ the absolute Galois group of $K$. Let $\bar{u} : G_K \longrightarrow \mathbb{F}_p$ a continuous additive ...
1
vote
0answers
132 views

how to prove that a Galois representation is exceptional?

A theorem of Deligne asserts that to cusp forms with Euler product, there is for each prime $\ell$ a Galois representation of $G(K_{\ell}/\mathbb{Q})$, where $K_{\ell}$ is the maximal abelian ...