The galois-representations tag has no usage guidance.

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### 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 ...

**25**

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**2**answers

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### What are the possible motivic Galois groups over $\mathbb Q$?

Let $E$ be a motive over $\mathbb Q$. (I should precise, that by a motive I mean
here a pure motive over $\mathbb Q$, with coefficients in $\mathbb Q$, that I see here as a conjectural object which ...

**24**

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**1**answer

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### Modern proof of Serre's open image theorem?

Let $E$ be an elliptic curve defined over a number field $K$ without complex multiplication. Serre's open image theorem (which appears in his book 'Abelian $l$-Adic Representations and Elliptic ...

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### Does the image of a p-adic Galois representation always lie in a finite extension?

I have been looking at Serre's conjecture and noticed that there are two conventions in the literature for a p-adic representation $\rho:\mbox{Gal}(\bar{\mathbb Q}/\mathbb Q)\to \mbox{GL}(n,V).$ In ...

**22**

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**1**answer

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### Fontaine-Mazur for GL_1

For any number field $K$, the Fontaine-Mazur conjecture predicts that any potentially semistable $p$-adic representation of the absolute Galois group $G_K$ of $K$ that is almost everywhere unramified ...

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**1**answer

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### Iwasawa main conjectures vs Bloch-Kato conjectures

Let $p$ be a prime, $K$ be a number field, $S$ a finite set of finite places of $K$ containing the set $S_p$ of places above $p$ and the places at infinity, $G:=G_{K,S}$ the Galois group of the ...

**21**

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### Introductory text on Galois representations

Could someone please recommend a good introductory text on Galois representations? In particular, something that might help with reading Serre's "Abelian l-Adic Representations and Elliptic Curves" ...

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**3**answers

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### 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 $G_{Q_p}$ ...

**21**

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### 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**

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**6**answers

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### 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 ...

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### Subgroups of GL(2,q)

I have been wondering about something for a while now, and the simplest incarnation of it is the following question:
Find a finite group that is not a subgroup of any $GL_2(q)$.
Here, $GL_2(q)$ ...

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### 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 ...

**19**

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**1**answer

696 views

### Status of conjectures in Serre's 1969 expose on Galois representations on l-adic cohomology

In
[S]: Serre, Jean-Pierre. Facteurs locaux des fonctions zeta des varietes algebriques (definitions et conjectures), Seminaire Delange-Pisot-Poitou, 1969-70
Serre presents nine conjectures ...

**18**

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**0**answers

613 views

### Eichler-Shimura over Totally Real Fields

By Eichler-Shimura over totally real fields I mean the following conjecture.
Conjecture. Let $K$ be a totally real field. Let $f$ be a Hilbert eigenform with rational eigenvalues, of parallel weight ...

**16**

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**1**answer

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### Are there Maass forms where the expected Galois representation is $\ell$-adic?

Recall that by theorems of Deligne and Deligne--Serre, there is the following dichotomy:
Modular forms on the upper half plane of level $N$ and weight $k\geq 2$ correspond to representations ...

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**1**answer

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### Fontaine's rings of periods

I've been trying lately to understand Fontaine's rings of periods, $B_{\mathrm{dR}}$, $B_{\mathrm{cris}}$, etc. However, I have a really hard time understanding and appreciating how to think about and ...

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**2**answers

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### $p$-adic Langlands correspondence

Basic question: Is it correct that the $p$-adic Langlands correspondence is known for $GL_2$ only over $Q_p$ but not other $p$-adic fields? If so, I would like to request some light to be shed on this ...

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**1**answer

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### If the tensor product of two representations are crystalline, are the original representations crystalline?

Let $K$ be a finite extension of the $p$-adic numbers. Suppose that $V$ and $W$ are two (finite dimensional, $p$-adic) continuous representations of $G_K$. Suppose that $V \otimes W$ is crystalline. ...

**15**

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**0**answers

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### Special values of Artin L-functions

This question might be naive and might carry the heuristic that we are living in the best possible world a little too far. If so, I appreciate being told so.
Background: Stark's conjecture interprets ...

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### Status of Fontaine-Mazur conjecture

In the language of Richard Taylor's 2004 (extended) ICM article (''Galois Representations'', Annales de la faculté des sciences de Toulouse (2004) Tome XIII, no. 1, 73-119), the conjecture is the ...

**14**

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**1**answer

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### Local-global principle for split extensions of Galois representations

I guess the following is well-known (and probably follows from Chebotarev's density theorem, but I'm not very comfortable with it):
Define some notation:
$K$ a global field,
$G$ the absolute Galois ...

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**1**answer

951 views

### Is there a Montgomery's conjecture for Dirichlet characters and Artin representations ?

Edit: as GH noticed, the way I tried to state Montgomery's conjecture is wrong. There were some mistakes in the references I used, which compounded with some mistakes of mine, gave a very poor post. ...

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### Representations attached to p-adic modular forms

A theorem of Gouvea and Hida (or rather a consequence of it) states that there exist a Galois representation attached to a $p$-adic eigenform $f$ provided the residual representation attached to a ...

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**1**answer

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### Galois representations attached to Maass form

So, how does one construct a galois representation from a Maass form?
For a modular cusp eigenform, I am familiar with the work of Eichler-Shimura, Deligne, Deligne-Serre, and realize these are ...

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**1**answer

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### 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 ...

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**2**answers

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### 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 $p$. ...

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### 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 ...

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### Elliptic curves and supercuspidal representations of conductor $p^2$

Let $E$ be an elliptic curve defined over $\mathbf{Q}$. Let $p \geq 5$ be a prime of additive reduction for $E$.
Let $f$ be the newform associated to $E$, and let $\pi$ be the irreducible admissible ...

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### 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 ...

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### What are the strongest conjectured uniform versions of Serre's Open Image Theorem?

This question concerns the uniform conjectured effective versions and generalizations of these two results of Serre on $\ell$-adic Galois representations $\rho_{E,\ell}$ associated to a non-CM ...

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### partition functions and Galois representations?

The recent answer's link to Ono's work makes me ask and wonder if his new results on partition functions tell something about Galois representations? (Hoping that question is not a case of this)

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### Period rings for Galois representations

I have some questions concerning period rings for Galois representations.
First, consider the case of $p$-adic representations of the absolute Galois group $G_K$, where $K$ denote a $p$-adic field. ...

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### Are there known cases of the Mumford–Tate conjecture that do not use Abelian varieties?

(For a formulation of the Mumford–Tate conjecture, see below.)
The question
As far as I know, all non-trivial known cases of the Mumford–Tate conjecture more or less depend on the Mumford–Tate ...

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### What is the classification of characters in $p$-adic Hodge theory?

Let $K$ be a $p$-adic field and $\chi : Gal_K \rightarrow \mathbb{Q}_p^\times$ be a character. I know that $\chi$ is Hodge-Tate of weight $0$ iff $\chi(I_K)$ is finite (by Sen's theory), and that it ...

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### Explicit Chebotarev and Langlands - irreducibility of X^5-X-1 mod primes

Is there an explicit infinite set of primes, modulo which $X^5 - X - 1$ is irreducible?
Since our polynomial's Galois group over $\mathbb{Q}$ is $S_5$, Chebotarev's density theorem implies that ...

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### Is there a “trianguline period ring”, or is one expected?

Consider a finite-dimensional $\mathbf{Q}_p$-vector space $V$ and a continuous representation $\rho : G_{\mathbf{Q}_p} \to \mathrm{GL}(V)$. Fontaine introduced various $\mathbf{Q}_p$-algebras with ...

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### To what extent are modular parametrizations expected to generalize?

By the Modularity Theorem (a.k.a. the Shimura--Taniyama--Weil Conjecture), if $E$ is an elliptic curve over $\textbf{Q}$ with conductor $N$, then there exists a “modular parametrization” $\psi: X_0(N) ...

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### Why doesn't functoriality immediately imply the modularity theorem?

Let $E/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, the prime indexed coefficients of its $L$-function agree with those of a weight $2$ cusp eigenform $f$ with integer coefficients. ...

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### Deligne's letter to Jean-Pierre Serre

I'm looking for another letter of Pierre Deligne, this time to Jean-Pierre Serre (from around 1974 I think), in which he proves that the Galois representation associated to a certain Hecke eigenform ...

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**1**answer

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### When is the image of a 2-dim l-adic representation associated to a modular form open

I know the following theorems by Serre:
1, The 2-dim l-adic representation associated to a non-CM elliptic curve is open.
2, The 2-dim l-adic representation associated the weight-12 cusp form ...

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### About isogeny theorem for elliptic curves

$K$ a number field, $G_K$ its Galois group, $E_1, E_2$ two elliptic curves defined over $K$. The isogeny theorem says that if for some prime number $\ell$, The Tate modules (tensored with ...

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### Best bounds toward Serre's uniformity conjecture

If $E$ is a non-CM elliptic curve over $Q$, then it is a famous theorem of Serre
that there is some integer $M(E)$ such that for any prime $\ell > M(E)$, the image of the
Galois representations ...

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### Galois action on one-dimensional quotients of l-adic cohomology

Let $A$ be an abelian variety of dimension $g$ over a number field $K$, and $\ell$ be a rational prime. Suppose that the Galois action on the $\ell$-adic cohomology $H^k(A, \mathbb{Z}_\ell) ...

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### Serre's Open Image Theorem Without Shafarevich's Theorem

In Abelian l-adic Representations and Elliptic Curves (1968), J. P. Serre showed that the adelic representation $$\rho_{E}\colon G_K \to \mathrm{GL}(\hat{\mathbb{Z}}^2)$$ associated to an elliptic ...

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**1**answer

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### Potential semi-stability of etale cohomology of etale covers.

Let $\mathbf{Q}_p$ denote the field of $p$-adic numbers.
Suppose that $K/\mathbf{Q}_p$ is a finite extension, and let $O_K$ denote the ring of integers of $K$. Suppose that $X$ is proper over $O_K$, ...

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### Is there literature on approaching semisimplicity of l-adic cohomology using vanishing cycles

Let $K$ be a finitely generated field, and $X/K$ a smooth proper variety. Let $G$ denote the absolute Galois group $\mathrm{Gal}(\bar{K}/K)$. Let $\ell$ be a prime different from $\mathrm{char}(K)$. A ...

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### Why does $H^i(X_{ét},\mathbb{Q}_p)$ have a Hodge-Tate structure?

Let $X$ be a variety over a $p$-adic field $K$.
Is there a simple or intuitive explanation of why the $G_K$ representation $H^i(X_{ét},\mathbb{Q}_p)$ is Hodge-Tate? More precisely, why do the powers ...

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### What is the etymology for the term conductor?

This is related to the previous question of how to define a conductor of an elliptic curve or a Galois representation.
What motivated the use of the word "conductor" in the first place?
A friend ...

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### how irregular can a $p$-adic Galois representation be?

Fix $p>0$ a rational prime, and $K$ an algebraic number field with Galois group $G_K:=Gal(\bar{\mathbb{Q}}/K) $. The Fontaine-Mazur conjecture predicts that if $\rho:G_K\rightarrow GL(V)$ is a ...

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### Why is there a weight 2 modular form congruent to any modular form

I got my copy of Computational Aspects of Modular Forms and Galois Representations in the mail yesterday. The goal of the book is "How one can compute in polynomial time the value of Ramanujan's tau ...