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14
votes
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
941 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
194 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
286 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
366 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 ...
4
votes
0answers
308 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
votes
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
282 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
295 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
vote
0answers
346 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
1k 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
304 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 ...
12
votes
1answer
978 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
653 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
228 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
164 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 ...
13
votes
0answers
511 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
241 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
344 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
votes
0answers
282 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
332 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
votes
0answers
352 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
324 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
215 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
531 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?
12
votes
0answers
4k views

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 ...
3
votes
1answer
760 views

Integral p-adic Hodge theory

Hello, Nowadays, I think we have some classification of integral structure in semistable representation via Liu's $(\varphi, \hat{G})$-modules or via Caruso's $(\varphi, \tau)$-modules. I must say ...
7
votes
2answers
849 views

P-adic representations

Hi, I am reading about p-adic representations from Fontaine's book which can be found at http://staff.ustc.edu.cn/~yiouyang/research.html. On page 145 where they prove Proposition 5.24 which is ...
2
votes
1answer
302 views

Terminology-history of p-adic representations

Where appears for the first time the term Hodge-Tate representation. Can i find somewhere explanation of the terminology Hodge-Tate, Derham etc. for representations and Fontaine's rings.
4
votes
1answer
280 views

What is the mod l monodromy of a generic trigonal curve?

For a hyperelliptic curve H, the mod 2 monodromy is smaller than $GSp_{2g}(F_2)$ -- since the two torsion of the Jacobian H is generated by differences of Weierstrass point, the monodromy of a generic ...
8
votes
1answer
333 views

Image of complex conjugation by modular representations in characteristic 2

The question I am going to ask looks well-known, and I even may have heard things about it (but since I used to be deaf to anything in characteristic 2, whatever I heard has never been recorded in my ...
3
votes
2answers
350 views

CM abelian varieties and potential good reduction

Let $F$ be a number field and $A$ an abelian variety over $F$. It is known that if $A$ has complex multiplication, then it has potentially good reduction everywhere, namely there exists a finite ...
10
votes
3answers
651 views

Cohomology of $SL_2(\mathbb{F}_p)$ acting on trace zero matrices over $\mathbb{F}_p$

I've been reading a paper by R.Ramakrishna about lifting Galois representations and at some point he uses the fact that a representation with big image has trivial first cohomology group. In other ...
4
votes
2answers
590 views

Elliptic curves over $\mathbf{Q}$ with isogenous mod $\ell$ reductions, for several $\ell$

Characteristic polynomials of Hecke operators $T_\ell$, with $\ell$ prime, acting on cusp forms $S_k$ of level one and weight $k$ "appear to be" squarefree (even irreducible!). This can be ...
16
votes
2answers
5k views

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" ...
3
votes
0answers
308 views

Which properties of p-adic representations can be recovered from (\varphi,\Gamma)-modules?

In Berger's "An introduction to the Theory of $p$-adic Representations", he mentions that due the the equivalence of categories between etale $(\varphi,\Gamma)$-modules and $p$-adic representations, ...
1
vote
1answer
394 views

Reference for “Gal represenations attached to CM eigenforms”

I seem to recall that the construction of Gal representations associated to eigenforms with CM was done much before the general cases due to Eichler-Shimura, Deligne-Serre and Deligne. Was this done ...
4
votes
0answers
272 views

Varieties corresponding to a given Galois representation

Given an $l$-adic Galois representation which is geometric in the sense of Fontaine-Mazur what can one say about the set of (isomorphism classes of) of varieties whose $l$-adic cohomologies the ...
2
votes
1answer
502 views

Serre's open image theorem for products of elliptic curves over function fields via specialization

In Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15, 259--331 (1972), Serre proved the following (Theorem 6 ′′, p. 325): Let $K$ be a number field and let ...
4
votes
2answers
479 views

Intersection of field extensions of torsion points of non-isogenous elliptic curves

Let $E$ and $E'$ be non-isogenous elliptic curves over a field $k$ (characteristic 0) such that $Gal(k(E[p^{\infty}])/k)=Gal(k(E'[p^{\infty}])/k) = SL_2(\mathbb{Z}_p)$ with $p \geq 5$ (where ...
5
votes
0answers
425 views

a naive question about p-adic local monodromy theorem

The question is about whether one can view the p-adic local monodromy theorem as the quasi-unipotence of some monodromy operator. it is known that the classical local monodromy theorem (i.e. for ...
12
votes
1answer
772 views

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 ...
8
votes
2answers
2k views

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 ...
8
votes
1answer
696 views

Motives from the fundamental group made nilpotent

I am reading the fascinating paper of Deligne on "le groupe fondamental de la droite projective moins trois points", and other stuffs related to anabelian geometry. This suggested the following ...
1
vote
1answer
545 views

Galois Cohomology maps

Let $\bar{\rho}$ be a residual ordinary and locally split galois representation associated to a weight $k$ level $1$ form (more specifically a mod $p$ companion form). In the sense of deformation ...
6
votes
0answers
373 views

dimensions of Galois representations appearing in the cohomology

Let's take $D$ to be $n^2$-dimensional central algebra with an involution $*$ over a CM field $E=FE_0$ (where F is totally real, and $E_0$ is imaginary quadratic). Define $$G(R) = \{ x \in D \otimes ...
5
votes
0answers
630 views

Reference for the Odd Dihedral Case of Artin's conjecture

The example that Matt Emerton cited here: prompted me to become interested in how one proves that odd two dimensional dihedral Galois representation are modular. This is the first case of the strong ...
2
votes
1answer
284 views

How do you calculate the Euler factors of the imprimitive symmetric square at primes with bad reduction?

The reference for this question is Coates and Schmidt, Iwasawa theory for the symmetric square. Let $G = \textrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}))$ and let $D_r \supseteq I_r$ be a ...
5
votes
1answer
521 views

Is the Galois x Hecke action on cohomology of Shimura varieties semi-simple?

Given a reductive group $G/\mathbf Q$ (+ additional data), and a compact open subgroup $K\subset G(\mathbf A^\infty)$, there is a standard construction that produces a Shimura variety $S$ and if we ...
1
vote
2answers
787 views

Decomposition of Artin L functions

The Dedekind zeta function of an abelian extension $E$ of $\mathbb{Q}$ factors as a product of Artin L function $L(s, \chi)$, where the product runs over all irreducible representations $\chi$ of ...