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21
votes
2answers
2k views

Elementary Aspects of Galois Deformation

Galois deformations are an important tool in Wiles' arsenal for proving FLT. Are there any more elementary aspects (I'm thinking of 1-dimensional Galois representations attached to number fields) ...
20
votes
3answers
1k views

One dimensional (phi,Gamma)-modules in char p

I would like to better understand the simplest case of the correspondence between Galois representations and (phi,Gamma)-modules. Namely, consider 1-dimensional Galois representations of $G_{Q_p}$ ...
17
votes
2answers
2k views

Number theory textbook based on the absolute Galois group?

I've just finished reading Ash and Gross's Fearless Symmetry, a wonderful little pop mathematics book on, among other things, Galois representations. The book made clear a very interesting ...
43
votes
6answers
3k views

What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$-adic field?

Let me stress that I am only interested in $p$-adic fields in this question, for reasons that will become clear later. Let me also stress that in some sense I am basically assuming that the reader ...
9
votes
2answers
1k views

Galois group of a product of polynomials

How can I compute the Galois group of the polynomial $fg\in K[x]$ assuming that I know the Galois groups of $f\in K[x]$ and $g\in K[x]$? Let's suppose for simplicity that the field $K$ is perfect.
12
votes
2answers
1k views

Galois representations attached to newforms

Suppose that $f$ is a weight $k$ newform for $\Gamma_1(N)$ with attached $p$-adic Galois representation $\rho_f$. Denote by $\rho_{f,p}$ the restriction of $\rho_f$ to a decomposition group at $p$. ...
6
votes
1answer
212 views

Cyclic extensions coming from E[p] \equiv F[p],

Let p be a prime and let K be a field containing the p'th roots of unity. Let E be an elliptic curve over K. We consider the the moduli problem $Y_E(p)$, which sends L to set of elliptic curves F/L, ...
20
votes
6answers
2k views

Where can I find a comprehensive list of equations for small genus modular curves?

Does there exist anywhere a comprehensive list of small genus modular curves $X_G$, for G a subgroup of GL(2,Z/(n))$? (say genus <= 2), together with equations? I'm particularly interested in genus ...
12
votes
1answer
1k views

The difficulties in proving modularity lifting theorems over non-totally real fields

First of all, let me apologize in advance for the terseness of this question. It seems that by now there are well-developed techniques (the "Taylor-Wiles-Kisin" method) for proving modularity lifting ...
12
votes
2answers
832 views

Obstructions to descend Galois invariant cycles

Let $X$ be a smooth projective variety over $F$, and $E/F$ - finite Galois extension. There is an extension of scalars map $CH^\*(X) \to CH^\*(X_E)$. The image lands in the Galois invariant part of ...
8
votes
2answers
668 views

number of irreducible representations over general fields

For a finite group, there are finitely many irreducible representations of complex numbers. What if the field is changed to some other fields? Like real numbers, p-adic field, finite field? In ...
10
votes
2answers
1k views

Galois theory and rational points on elliptic curves

I am in search of a concrete example [a concrete elliptic curve in Weierstrass form] of how Galois theory helps to find rational points on an elliptic curve. Chapter VI of Silverman and Tate discusses ...
7
votes
2answers
976 views

Free subquotient of Galois representations coming from Hida theory

Let $\mathbf{T}$ be the reduced nearly ordinary Hecke algebra of level $N$ of Hida theory for $\operatorname{GL}_{2}$ over $\mathbb{Q}$ (or more generally over a totally real field $F$). Then ...
4
votes
1answer
523 views

A name for primes where residual Galois representations are reducible

Let $\overline{\rho}_{\Delta,\ell}$ be the mod-$\ell$ representation associated to Ramanujan's $\Delta$-function. It is well-known that (the semisimplification of) this representation is reducible ...
10
votes
3answers
837 views

Motivation for uniform surjectivity of mod l representations associated to elliptic curves

Background Let $E$ be an elliptic curve over $\mathbb{Q}$ and let $G_{\mathbb{Q}}$ be the absolute Galois group $Aut(\overline{\mathbb{Q}})$. For any positive integer $n$ the $n$-torsion subgroup ...
9
votes
1answer
377 views

Level raising by prime powers

Suppose $f$ is a weight $2$ level $N$ cusp form. When can we realize the mod-$\ell$ representation of $f$ in a form of weight $2$ and level $Np^3$, where $p$ is some prime not dividing $N$? I assume ...
3
votes
1answer
635 views

An inverse problem: Number fields attached to elliptic curves over Q

If I understand FC's remark under the post "Very strong multiplicity one for Hecke eigenforms," in the course of Faltings's proof of the Tate conjecture, Faltings proves the following statement: let ...
5
votes
1answer
893 views

Learning about Galois representations

My goal was to learn about l-adic representations on some example — I'm a newbie in these topics. Thus take pt = Spec F_q, ...
8
votes
3answers
789 views

Does Ribet's level lowering theorem hold for prime powers?

I often use the following theorem (that one can state more generally) in my research. Let E/Q be an elliptic curve of conductor N corresponding to a modular form f(E), l a prime of good or ...