The tag has no wiki summary.

learn more… | top users | synonyms

6
votes
0answers
354 views

Benedict Gross's paper on companion forms

In the page 458 of his paper(A tameness criterion for Galois representations associated to modular forms), Gross wrote the following "A detailed analysis of $U_p(Af)+V_p(<p>f)$ shows that it ...
5
votes
1answer
145 views

Pro-$l$ Sylow action in a primitive representation of inertia over $\overline{\mathbb{F}}_l$

Let $K$ be a nonarchimedean local field of residue characteristic $p \neq l$ and let $I_K$ be the inertia subgroup of its absolute Galois group. Let $V$ an irreducible representation of $I_K$ over ...
5
votes
0answers
238 views

Which de Rham representations are trianguline?

Let $K/\mathbf{Q}_p$ be a finite extension, and let $V$ be an $n$-dimensional $\overline{\mathbf{Q}_p}$-vector space with a continuous action of $G_K$. Suppose $V$ is de Rham, so potentially ...
8
votes
0answers
267 views

Galois action on $E_n$-operads

Let $E_n$ be the little $n$-cubes operad which acts on $n$-fold loop spaces (up to group completion, an $E_n$-action is precisely the data needed to perform an $n$-fold delooping). I am looking for ...
6
votes
1answer
197 views

$(\varphi, \Gamma)$-modules of finite height

Maybe the answer to my question is obvious. Let $p$ be a prime $\geq 3$. Let $D$ be an étale $(\varphi, \Gamma)$-module over $A_{\mathbb{Q}_p} = \{ \sum_{n \in \mathbb{Z}} a_n X^n \, \vert \, a_n \in ...
10
votes
0answers
408 views

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 ...
12
votes
1answer
349 views

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 ...
4
votes
1answer
199 views

$(\varphi, \Gamma)$-module of dimension 2 modulo $p$

Let $p$ be a prime number $\geq 3$. Let $V$ be a representation of $Gal(\bar{\mathbb{Q}}_p/ \mathbb{Q}_p)$ with coefficients in $\mathbb{F}_p$. Assume $V$ is a non-split extension of two characters ...
6
votes
2answers
594 views

Peu or très ramifiée extension

Let $p$ be a prime number. Let $\mathbb{F}$ be a finite extension of $\mathbb{F}_p$. Let $\omega$ be the mod $p$ cyclotomic character and let $V$ be a representation of $G_{p} = Gal(\bar{\mathbb{Q}}_p ...
4
votes
2answers
225 views

Generalization of Kummer isomorphism?

This is a question I asked on math.stackexchange without success. 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 ...
3
votes
1answer
236 views

$\ell$-conductor of a two-dimensional $\ell$-adic Galois representation

Let $\ell$ be a prime number, denote by $K_\ell$ the maximal algebraic extension of $\Bbb{Q}$ ramified only at $\ell$. Let $f = \sum a_n q^n$ be a Hecke eigenform of level $1$ with integer ...
2
votes
0answers
144 views

Local components of quaternionic modular forms

Let $D$ be a totally definite quaternion algebra over a totally real number field $F$. Let $U$ be an open compact subgroup of $D(\mathbb{A}_F)^\times$, maximal compact almost everywhere. Consider ...
8
votes
1answer
316 views

Extensions of Galois representations

Let $G=Gal(\bar{\mathbb Q}/{\mathbb Q})$ be the absolute Galois group of the rationals. Fix two continuous group homomorphisms $\alpha,\beta: G\to {\mathbb Q}_l^\times$, where $l$ is a prime and ...
2
votes
1answer
233 views

Galois deformations with Panchiskin condition

Let $L/\mathbf{Q}_p$ be a finite extension and we consider a fixed $L$-linear representation $V$ of the absolute Galois group $G:=\operatorname{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$. Assume that ...
2
votes
0answers
119 views

Residue fields of attached to coefficients of modular forms

Let $f = \sum_n a_n q^n$ be a cuspidal newform of some weight and level. Here I want to view the $a_n$ of $p$-adic numbers (by embedding $\overline{{\bf Q}}_p$ in ${\bf C}$ in some way). Let ...
3
votes
2answers
414 views

Field generated by the Fourier coefficients of a modular form

Let $f = \sum_n a_n q^n$ be a cuspidal newform of weight $k$ on $\Gamma_0(N)$ for some $N$. Let $K_f$ be the number field generated by the $a_q$ as $q$ runs over all primes. My question: if we ...
9
votes
2answers
704 views

Effective Chebotarev without Artin's conjecture

Iwaniec and Kowalski, in their famous book Analytic Number Theory states a strong form of the effective Chebotarev density theorem page 143, and prove it assuming both GRH for Artin's $L$-function and ...
9
votes
2answers
519 views

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 ...
0
votes
0answers
325 views

Local Langlands conjecture for GL(2)

Let $F_v$ be a local field. Let $\sigma_v$ be a two-dimensional representation of $Gal(\overline{F}_v:F_v) \rtimes W_{F_v}$. Now, there exists an infinite-dimensional representation $\pi_v$ of ...
1
vote
1answer
215 views

Finite Flat Group Schemes for Modular Forms of Higher Weight

Let $f$ be a newform (normalized, cuspidal) of weight $k \ge 2$. Then for a prime $\ell$ there is an $\ell$-adic Galois representation associated to $f$. If $k=2$, this comes from an abelian variety, ...
3
votes
0answers
88 views

Decompositions of representations of pro-p groups

Let $P$ be a pro-p group. Assume that there is a filtration of $P$ by normal subgroups $P_i$ such that $P_0=P$ and $P_{i+1} < P_i(i\in\mathbb N)$. Let $V$ be an $l$-adic representation of $P$, ...
21
votes
1answer
736 views

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 ...
4
votes
0answers
104 views

Deformation rings and change of group

Let $f : G' \subset G$ be an injection between profinite groups such that $G'$ is normal in $G$ (typical situation which I deal with : $G$ the absolute Galois group of a local field, $G'$ an open ...
3
votes
1answer
203 views

An abelian Hodge-Tate representation lands in a torus

Bogomolov's paper "Sur l'algébricité des représentations l-adiques" proves that the image of the $\ell$-adic Galois representation associated to an abelian variety over a number field $K$ is open in ...
5
votes
0answers
465 views

Grothendieck monodromy theorem for l-adic sheaves

Hi, Suppose that $F$ is a local field, $G_F$ its Galois group, $I$ the inertia subgroup, $k$ its residue field. Let $X$ be a finite type scheme over $k$. Let $C$ be a constructible $l$-adic sheaf on ...
0
votes
0answers
144 views

Uniqueness of decomposition of completely reducible representations

Let $X$ be a smooth, separated scheme of finite type over $\mathbb{F}_q$ where $q=p^r$ for some $r>0$. Let $gcd(l,p)=1, \rho:W(X) \to GL_r(\mathbb{Q}_l)$ be a Weil representation which is ...
3
votes
0answers
199 views

The operator \boxtimes and \boxplus in automorphic representations

Given two automorphic representations $\pi_1, \pi_2$ of $GL_2(\mathbb A_Q)$ and $GL_3(\mathbb A_Q)$ respectively. Let $\pi_i =\otimes_v \pi_{i, v}$. Now, for each $v$, let $\pi_{1, v}\boxtimes ...
7
votes
2answers
540 views

Periods for 2-variable p-adic L-functions

Hi all, I am sorry to ask a stupid question but I am really confused right now. Kitagawa-Mazur constructed a $2$-variable p-adic L-function attached to Hida families of modular forms. For their ...
9
votes
2answers
566 views

What is a(n algebro-geometric) family of modular forms?

We know that a family of elliptic curves is a morphism of schemes $f:X \to Y$ such that the fiber of every point of $Y$ is an elliptic curve (and we usually require the morphism to be smooth, proper, ...
4
votes
1answer
514 views

Should the etale cohomology of a smooth projective variety (over rationals) be semi-simple; why?

$\DeclareMathOperator{\char}{char}\DeclareMathOperator{\gal}{Gal}$ Let $P$ be a smooth projective variety over a field $K$ (one may certainly assume that $K$ is perfect; the case $K=\mathbb{Q}$ ...
1
vote
1answer
356 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
vote
0answers
189 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
votes
0answers
187 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
vote
1answer
290 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
votes
3answers
349 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
votes
1answer
908 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
votes
0answers
208 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 ...
0
votes
0answers
55 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
votes
1answer
249 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
votes
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
votes
2answers
372 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
votes
1answer
395 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
326 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
436 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
votes
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
votes
1answer
859 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 ...
19
votes
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
409 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 ...
3
votes
1answer
214 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
545 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 ...