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23 views

Induced representations of compact groups

I am taking a seminar that follows Serre's book "Linear Representations of Finite Groups", and I am preparing a talk on Chapter 7 on induced representations (Frobenius reciprocity, Mackey's formula ...
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1answer
81 views

Cube root of discriminant of elliptic curve

Let $E/K$ be an elliptic curve over a field $K$, with discriminant $\Delta$. Then the polynomial $x^3-\Delta$ has a root (and hence all roots since Galois) in $K(E[3])$; this can be shown laboriously ...
2
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1answer
63 views

Does the supposed to exist functor considered in Langlands program bear a peculiar name?

I'm trying to figure out what a very rough sketch of the Langlands program could be. From what I (think I) understand, objects called reductive algebraic groups together with related so-called ...
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0answers
24 views

Some article about Galois representation

I have heard that the Galois Representation associated to a modular form which came form an elliptic curve with CM type has a small image.Could anybody tell me some article about this? I have heard ...
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61 views

On a certain representation of the Galois group of $X^n-a$ from Lang's Algebra

I am having trouble understand theorem $9.4$ of Chapter $6$ of Lang's Algebra (pg. 300-301). The setup is a we have a field $k$ of characteristic not dividing $n$. We know that the splitting field ...
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37 views

Why is the image of a $\pmod p$ Galois representation finite?

Let $\overline{\mathbb{F}_q}$ be the algebraic closure of the finite field on $q=p^r$ elements, and $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ the absolute Galois group with the profinite ...
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37 views

locally algebraic representations

Let $K$ be a number field. Consider $$ \rho_{\ell}: \mathrm{Gal}(\bar K/K) \longrightarrow \mathrm{GL}(V)$$ an $\ell$-adic Galois representation. Assume it is semi-simple rational and abelian. Is ...
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37 views

Conjugacy Class in Galois Representations

Let $G=Gal(\bar K/K)$ be the absolute Galois group of a number field $K$. Let $v$ be a finite place of $K$ and $w$ a place of $\bar K$ extending $v$. Take an $\ell$-adic representation $$ \rho: G ...
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1answer
102 views

How does Galois group acts on etale cohomology?

I know this may be a trivial question, but I can't find the answer on, for example, Milne's online notes and Danilov's Cohomology of Algebraic Varieties. Suppose $K$ is a number field (say), ...
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64 views

Tate curve and action of inertia group

I read the answers to this question Clarifying a comment of Serre. However I miss a passage of the second answer and since I can't comment there I have should post a new question. I don't understand ...
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0answers
30 views

element of an $\ell$-adic Galois representation with certain eigenvalues.

Let $\mathscr{G}=Gal(\bar{\mathbb{Q}} / \mathbb{Q})$, $E$ an elliptic curve over $\mathbb{Q}$, and consider the $\ell$-adic representation $$ \varphi_{\ell}: \mathscr{G} \longrightarrow ...
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1answer
91 views

References about moduli space of abelian varieties with level structure

In the course of one of my research project, I have been advised to try to have a look to "Moduli Space of Abelian Varieties with Level Structure". I am interested in references where this topic is ...
2
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1answer
44 views

Abelian semi-simplification representation

Let $K$ be a number field with $G=\mathrm{Gal}(\bar K/K)$. Consider the representation of $G$ with value in a $\mathbb{F}_p$-vector space $V$ of dimension 2 $$ \rho : G \longrightarrow ...
3
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1answer
45 views

Galois Representation with $D_{10}$ image

I want to construct an explicit elliptic curve $E$ over a number field $K$ such that $Gal(K(E[l])/K) \cong D_{10}$ where $D_{10}$ is the dihedral group of order 10 and $l$ is a prime number. ...
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1answer
252 views

rationality of $\ell$-adic representation attached to an elliptic curves

Let $E$ be an elliptic curves defined over a number field $K$. Consider the $\ell$-adic representation attached to $E$ $$ \rho_{\ell}:\mathrm{Gal}(\overline{K}/K) \longrightarrow ...
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44 views

Representations of algebraic group ($S_{\mathfrak{m}}$)

I'm studying Serre's book "Abelian $\ell$-adic Representations and Elliptic Curves" and in chapter II $\S$2.4 we have this proposition: Consider $v$ a finite place of $K$ and $F_v \in Gal(K^{ab}/K)$ ...
3
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2answers
58 views

Continuity of Galois representations from cohomology

One of the most standard way to construct Galois representations is the geometric way: one starts from a variety $X$ defined over $\bf Q$ say; the Galois group acts on ${\overline X}:= X \times {\rm ...
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40 views

Connection between sgn character and the Legendre symbol

Today, while I was lecturing on the Legendre symbol, I realized that the phenomenon: "the product of two non-squares is a square" isn't so foreign. For example, for $\mathbb R^\times$, the squares are ...
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76 views

The Frobenius Trace for an elliptic curve

Let E be an elliptic curve defined over $\mathbb{Q}$ (coeffs. there), and consider its $n-$torsion points in $\mathbb{C}$, $E(\mathbb{C})_{\text{tors}}[n]$. We know this group is isomorphic to ...
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1answer
33 views

The character of a newform

In "Modular Forms and Fermat's Last Theorem" Chp.1, p.7 Glenn Stevens talks about the character of a weight 2 newform $\epsilon$ in relation to the characteristic polynomial of a Frobenius element. ...
3
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2answers
104 views

In a vector space over a finite field, can the orbit of a point under matrix multiplication have dependent subsets?

Let $\mathbb{F}_q^n$ be the vector space of dimension n over the finite field of order q, $\vec{v}$ a vector in the space, and $M$ an invertible $n \times n$ matrix over $\mathbb{F}_q$. We know that ...
3
votes
1answer
64 views

Weil pairing, cyclotomic field in division field and determinant map for ell curve and abelian variety

Question 1: If $E/K$ is an elliptic curve defined over a number field $K$, then the Weil Pairing gives me that $K(\mu_n) \subseteq K(E[n](\bar{K}))$. If i identify $Gal(K(\mu_n)/K)$ with a subgroup ...
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1answer
44 views

Definition of a “modular Galois representation”

I am trying to pin down a definition for a $n$-dimensional modular Galois representation $$\rho : \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{GL}_n(A).$$ I am just looking for ...
5
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1answer
118 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 ...
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1answer
110 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. ...
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1answer
91 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 ...
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votes
2answers
101 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
58 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), ...
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1answer
335 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 ...
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0answers
72 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 ...
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1answer
52 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 ...
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2answers
171 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 ...
2
votes
1answer
56 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*} ...
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1answer
44 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 ...
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1answer
97 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 ...
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1answer
85 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 ...
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1answer
398 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 ...
5
votes
1answer
97 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 ...
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1answer
211 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 ...
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1answer
63 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 ...
2
votes
1answer
63 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 ...
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1answer
240 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
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2answers
185 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 ...
6
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2answers
218 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 $\Bbb C$. ...
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1answer
59 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
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1answer
123 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
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1answer
70 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 ...
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1answer
81 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 ...
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1answer
199 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
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1answer
349 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 ...