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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.

Many objects that arise in number theory are naturally Galois representations. For example, if $L$ is a Galois extension of a number field $K$, the ring of integers $O_L$ of $L$ is a Galois module over $O_K$ for the Galois group of $L/K$ (see Hilbert–Speiser theorem). If $K$ is a local field, the multiplicative group of its separable closure is a module for the absolute Galois group of $K$ and its study leads to local class field theory. For global class field theory, the union of the idele class groups of all finite separable extensions of $K$ is used instead (Wikipedia).

307 questions
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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 representations of $G_K$ for p-adic fields $K$,…
Joachim
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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 $$ \rho: \text{Gal}(\overline{\textbf{Q}}/\textbf{Q}) \to GL_n(\overline{\textbf{Q}}_\ell) $$ "comes from geometry," that is,…
Miles Lake
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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 $\Bbb C$. Then complex conjugation in $\Bbb C$ restricts to a…
Dylan Yott
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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}:{\rm…
The Thin Whistler
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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 case where the image of Galois is the normalizer…
Tony
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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 through solving the 3-division polynomial (a…
Matt B
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What is a semistable representation?

This feels like a silly question, but it keeps coming up in seminars and even after much searching I still have no idea what it means. I would like to see a definition and some examples of things that are and aren't semistable, as well as why this…
ggg
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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), $\overline K$ its algebraic closure,…
Jz Pan
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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 $p$ cyclotomic character. We have a nice description of…
user33624
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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 Galois extension which led me to consider following…
tghyde
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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 coefficients. The $L$-function $L(f,s)$ admits an Euler…
Keenan Kidwell
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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 irreducible continuous and finite dimensional…
user33624
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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 $l$-adic Lie algebra stuff, or a general…
Bob G
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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 $E$ $$ \rho_{\ell}:\mathrm{Gal}(\overline{K}/K) \longrightarrow \mathrm{Aut}(V_{\ell})$$ With $\overline{K}$ an algebraic closure of…
user253407
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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 the Galois group of the maximal cyclotomic…
Dylan Yott
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