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1
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1answer
409 views

Hodge-Tate weights of etale cohomology

Let $K/\mathbb Q_p$ be a local field, $X/K$ a proper scheme with semi-stable reduction. Question: What is the possible range of Hodge-Tate weights of the etale cohomology $H^i(X_{\overline K}, ...
1
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0answers
200 views

Extending systems of l-adic representations to other l

I'm asking this not because I have an idea how one might approach it, but because it seems natural and inherently interesting. Let $K$ be a number field, $G_K$ its absolute Galois group, and ...
4
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0answers
207 views

Do infinite and ramified local factors of the Dedekind zeta function of a tame number field characterize its local root numbers?

Let say you have two number fields, that are tamely ramified, and suppose that the $p$-part of their Dedekind zeta functions coincide for all prime $p$ which is ramified in either field. Suppose ...
1
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1answer
299 views

Tate Conjecture on decomposition of motives(?)

I apologize for the title. I myself conined it... I am referring to Conjecture 1.2 (page 7) of Richard Taylor's paper Galois representations (Annales de la Faculte des Sciences de Toulouse 13 (2004) ...
2
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3answers
377 views

Useful notion of unramified Galois representation

Let $\mathbf C(t)$ be the field of rational functions and let $\overline{\mathbf C(t)}$ be an algebraic closure. Let $G$ be the Galois group of $\overline {\mathbf C(t)}$ over $\mathbf C(t)$. Let ...
14
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1answer
935 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. ...
3
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0answers
218 views

modular forms with lots of companion forms

Let $f$ be a modular form (say of weight 2) and let $S$ be the set of primes $p$, the reductions modulo $p$ of $\rho_{f,p}$ restricted to $G_p$ splits. Equivalently, $S$ is the set of prime $p$ such ...
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0answers
57 views

Commuting invariants and duals of C_p vector spaces

Let $K$ be a field complete with respect to some discrete valuation, with perfect residue field of characteristic $p$. Let $\mathbb{C}_p$ be the completion of an algebraic closure of $K$, and set $G_K ...
6
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1answer
256 views

The existence of an elliptic curve with a specific Galois representation induced by a character

In Kevin Buzzard's survey article on potential modularity Buzzard writes: Let us say that we have an elliptic curve $E$ over a totally real field $F$, and we want to prove that $E$ is ...
12
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0answers
1k views

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. ...
8
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2answers
384 views

Can one ignore primes lying over $l$ in the Fontaine-Mazur conjecture? Counterexamples?

The Fontaine-Mazur conjecture predicts that an $l$-adic Galois representation of a number field is 'geometric' if it is unramified outside a finite set of primes and is De Rham for primes lying over ...
5
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1answer
421 views

Level lowering for weight 1 forms

I'm interested in knowing to what extent is level lowering known to hold in weight 1. Specifically, let's say I have an eigenform $f$ in $S_1(N,\chi)$ and a prime $p$ which doesn't divide $N$. Let's ...
6
votes
3answers
338 views

Example of a diophantine application of an open image theorem

I'm an applied model theorist, and open image theorems are important in the mathematical structures I study (they limit the number of types of elements being realised, and therefore keep things model ...
3
votes
3answers
442 views

Another question related to the isogeny theorem for elliptic curves

I was reading the following question: About isogeny theorem for elliptic curves and was interested in the following statement at the end of Torsten Ekedahl's answer: "Note also that the situation is ...
9
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5answers
1k views

The significance of modularity for all Galois representations

On pg. 1 of the slides of a talk, Henri Darmon wrote: Question: What is an interesting Diophantine equation? A “working definition”. A Diophantine equation is interesting if it reveals or ...
13
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1answer
913 views

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 ...
22
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1answer
2k views

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 ...
4
votes
1answer
453 views

Duals and Tate twists of Galois representations attached to modular forms.

Let $f$ be a normalized newform of weight $k\geq 2$, $p$ a prime, and $V$ the associated Galois representation (with coefficients in a finite extension $K$ of $\mathbf{Q}_p$ with ring of integers ...
5
votes
2answers
283 views

Is there a semisimple $\mathbf{Q}_\ell$-representation of $G_F$ ramified at an infinite set of places?

See http://math.uni.lu/~wiese/galois/Boeckle-Luxemburg-Notes.pdf, Theorem 1.4(a): Is there an example of a semisimple $\mathbf{Q}_\ell$-representation $V$ of $G_F$ ($F$ a global field) ramified at a ...
6
votes
2answers
580 views

(phi, Gamma) module of ordinary elliptic curve

Suppose $E$ is an elliptic curve over $\mathbb{Q}_p$ with good ordinary reduction. Can someone please tell me how to compute the associated $(\phi,\Gamma)$-module of the Tate module of $E$, or give ...
4
votes
1answer
2k views

Why are Galois Representations so important in Number theory ?

Dear everyone, Motivation : From the past few days, I have been reading about the Galois Representations . I was really amazed to see that every seminal idea in the theory of elliptic curves have ...
1
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0answers
139 views

Construction of RM abelian variety from eigenform

Let $f$ be a normalized eigenform of weight $2$ level $N$. If the Fourier coefficients of $f$ generate a totally real field $F$, then we associate to $f$ a system of $\ell$-adic Galois representations ...
11
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2answers
893 views

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 ...
5
votes
1answer
387 views

Are coefficients of Maass forms of eigenvalue 1/4 known to be algebraic?

I would really like to know whether the following famous conjecture has been solved. I've read in a few places that it has been solved, but I have been unable to find a reference. I do know that ...
3
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0answers
351 views

local deformation rings and Hecke algebras

Let $\bar{\rho}_p$ be a two-dimensional irreducible local Galois representation of $Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p)$ on a $k$-vector space, where $k$ is a finite extension of $\mathbb{F}_p$. ...
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3answers
1k views

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 ...
4
votes
2answers
954 views

non-continuous inverse Galois problem

Let $G=Gal(\bar{\mathbf{Q}}/\mathbf{Q})$ be the absolute Galois group over $\mathbf{Q}$. Q1: Is it possible to find a (necessarily non-closed) normal subgroup $K\leq G$ such that $G/K$ is free of ...
1
vote
0answers
198 views

Where can I find a copy of Serre's Cours au college de France 1985-1986?

Hi, I was wondering: where might I be able to find a copy of this work online? And are there any other resources for the proof of the open image theorem for abelian varieties with endomorphism ring ...
2
votes
1answer
321 views

1-dimensional semi-stable Galois representations with coefficients

For any p-adic local field K, all 1-dim semi-stable Galois repn: $G_K \to Q_p^{*}$ are just $Q_p(n)\otimes \mu$, where $Q_p(n)$ is the Tate twist of cyclotomic character, and $\mu$ an unramified ...
3
votes
2answers
379 views

Quotients of Tate modules

Let $p$ be a prime number, let $K$ denote a finite extension $\mathbb{Q}_{p}$ and let $\overline{K}$ be an algebraic closure of $K$. Let $A$ be an ellitpic curve over $K$ and denote by $T_{p}A$ its ...
10
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1answer
423 views

Reference for: CM Hilbert Modular forms arise from Hecke characters

For classical modular forms, the correspondence between the form having CM by an imaginary quadratic field $K$ and it being induced from a Hecke character on $K$ is well-known. (Ribet's paper is a ...
10
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2answers
1k views

“Purely local” proof of local Langlands

As from this website http://math.uchicago.edu/~lxiao/workshop_site/ My question is: What does it mean by "purely local"? Also, I heard about this phrase "purely local" in other problems as well, ...
4
votes
1answer
295 views

semisimplicity of automorphic Galois representations

Is it known that the Galois representation constructed by Harris and Taylor in their book is semisimple? I can't see this proven in the book, but on the other hand, everywhere else the representation ...
2
votes
1answer
304 views

$l$-adic representations from Shimura curves

This question may be kind of vague. And we use the same notations as in Carayol's papers: H. Carayol, Sur les représentations l-adiques associées aux formes modulaires de Hilbert; H. Carayol, Sur la ...
1
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0answers
350 views

about Deligne's 1969 paper

First one: In his paper he says $GL_2(A^{f})$could act on the direct limit of the first cohomology groups with compact support $ H^1_c(M_n^{an},Sym^{k}(R^1f_{*}(Q)))$ with $n\rightarrow\infty$. Does ...
20
votes
3answers
2k views

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, $GL_2(q)$ ...
7
votes
0answers
330 views

Tameness criterion in the reducible case

Dear MO, This is a follow up to a previous question here in MO, but I will make this question self-contained for convenience. Those already familiar with the following paper [G] by Gross can safely ...
13
votes
1answer
1k views

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 ...
7
votes
1answer
757 views

Is there an integral version of Faltings' isomorphism in p-adic Hodge theory between etale and Hodge cohomologies

Let $K$ be a $p$-adic field, that is a complete discrete valuation ring of characteristic $0$ with a perfect residue field $k$ of characteristic $p > 0$ (to simplify one could also take $K$ to be a ...
4
votes
1answer
249 views

Explicitly computing D(V)^{psi=1} for (phi,Gamma)-modules

In the paper Construction of some families of 2-dimensional crystalline representations by Berger-Li-Zhu, a very explicit description is given for the $(\phi,\Gamma)$-module attached to some ...
2
votes
1answer
189 views

differential structures on unit-root Frobenius modules.

Let $\mathcal{E}$ be the ring of series $f(T)=\sum_{n \in \mathbb{Z}} a_nT^n, \; a_n \in \mathbb{Q}_p$ such that the $\{a_n\}$ are bounded and $a_{-n}\rightarrow 0$ as $n \rightarrow +\infty$. This is ...
14
votes
0answers
621 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 ...
0
votes
1answer
249 views

On the image of the residual representation attached to a CM form

It is a classical result of Ribet that if an eigenform has CM the its residual projective image is "small" (cyclic or dihedral.) Is the converse true, i.e, if f is a form whose associated residual Gal ...
3
votes
1answer
365 views

What is the relation of the absolute Galois group and classical profinite groups?

Consider the absolute Galois group $G = \mathrm{Gal}(\overline{\mathbb{Q}}: \mathbb{Q})$ and $G_p = \mathrm{Gal}(\overline{\mathbb{Q}_p}: \mathbb{Q}_p)$. Abelian class field theory gives us for the ...
4
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0answers
290 views

Local splitting of modular Galois representations as $p$ varies

If $f$ is a classical eigenform of weight $\geq 2$ and ordinary at distinct, odd primes $p$ and $q$ which do not divide the level is it true that the restriction (as a $q$-adic representation) ...
3
votes
2answers
338 views

Number of modular lifts with prescribed parameters

Let $\bar{\rho} : Gal(\bar{\mathbb{Q}}/ \mathbb{Q} ) \rightarrow GL_2(\bar{\mathbb{F}}_p)$ be an odd, irreducible Galois representation mod $p$ which is unramified outside $S$, where $S$ is a finite ...
4
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0answers
373 views

Mazur's relative deformation functors

In his paper "Deformation Theory of Galois Representations" in the book "Modular Forms of Fermat Last Theorem", Mazur considers more general deformation functors than in his earlier paper in "Galois ...
5
votes
1answer
341 views

Can a p-adic representation and its twist by a non-crystalline character both have nontrivial $D_{cris}$?

For a continuous irreducible representation $\rho: G_{\mathbb{Q}_p}\rightarrow GL_n(\overline{\mathbb{Q}_p})$, is it possible for both $D_{cris}(\rho)$ and $D_{cris}(\chi\otimes\rho)$ to be nonzero, ...
1
vote
1answer
218 views

Dimension of fixed points of Galois group actions

I have a question about fixed points of Galois group actions. I am hoping that this is easy for the experts. Let $k$ be a field of characteristic $0$. Let $K$ be a finite Galois extension of $k$ ...
6
votes
1answer
588 views

Mumford-Tate group and Galois representations

Could someone please point me towards a proof of why the image of a Galois representation on the Tate-module of an abelian variety is limited by its Mumford-Tate group?