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4
votes
2answers
483 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
442 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
786 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
738 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
554 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
382 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
649 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
297 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
534 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
819 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 ...
4
votes
1answer
730 views

Eichler-Shimura for Shimura curves

Hi, What is the statement of the Eichler-Shimura relation for Shimura curves? And where can one find a proof? Thanks
3
votes
0answers
455 views

choice of local system in Deligne's construction of $l$-adic Galois representations

Hello, Deligne famously constructed $l$-adic representations of $G_\mathbf Q = Gal(\overline{\mathbf Q}/\mathbf Q)$ starting form cusp modular forms of weight $k$ by looking inside the cohomology ...
0
votes
1answer
389 views

Poitou-Tate dualities for Galois representations into power series rings?

Suppose $K$ is a finite extension of $\mathbf{Q}_p$, $A=K[[T_1,\dots,T_n]]$, $V$ a finite-rank free $A$-module, and $\rho:G_{\mathbf{Q}} \to \mathrm{GL}(V)$ a continuous Galois representation. Are ...
1
vote
1answer
793 views

Rapoport-Zink proof of purity of monodromy

Hi, Does anyone know if the article "Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindene Zyklen in ungleicher Charakteristik", INvent. Math, 68 (1980) by ...
9
votes
1answer
876 views

Carayol via the trace formula

Hi, Is there a proof of the result that Carayol proves in "Sur les representations l-adiques..." using the Langlands-Kottwitz method of comparing the Lefschetz trace formula and the Selberg trace ...
9
votes
1answer
708 views

Tamagawa numbers of crystalline Galois representations

This is a followup to this question. Let $p \ge 3$ be prime, and let $V$ be a crystalline 2-dimensional representation of $G_{\mathbb{Q}_p}$ and $T$ a lattice in $V$. I'm going to assume just about ...
3
votes
3answers
784 views

Companion forms

What is the best known result concerning the existence of companion forms for classical modular forms? Gross' tameness criterion paper is always mentioned with a "unchecked compatibility" caveat? ...
2
votes
2answers
332 views

Why does $H^1(G_p/I_p,\mathbb{F}(\delta\epsilon^{-1}))$ vanish?

For a finite field $\mathbb{F}$ of char $p$ $\geq 2$, let $f$ be a normalised eigenform of weight $k\geq 2$ that is ordinary at $p$ and $\overline{\rho}_f$ be the mod $p$ Gal representation attached ...
8
votes
1answer
514 views

Do Tamagawa numbers of Galois representations stabilise in the cyclotomic tower?

Suppose $T$ is a free finite rank $\mathbb{Z}_p$-module with a continuous action of $\operatorname{Gal}(\overline{K} / K)$, where $K$ is a number field. There is a definition of local Tamagawa numbers ...
2
votes
1answer
368 views

How can we extend Galois representations ?

Let $F$ be a number field and $E/F$ a Galois extension. Suppose we have a representation $\rho_E : Gal(\overline{F}/E) \rightarrow GL_n(\overline{Q}_p)$. My question is : what are sufficiant ...
10
votes
0answers
203 views

Mod m versions of the toric part of Tate modules

Let $A$ be a polarized abelian variety over a local field $K$ with residue characteristic $p$. In the course of proving that a polarized abelian variety $A/K$ has semi-stable reduction iff for all ...
7
votes
0answers
601 views

Semistable Elliptic Curves and irreducible Galois representations

I am interested in the set of number fields $K$ having the property that for \emph{any semistable} elliptic curve $E$ defined over $K$, there exists a constant $c(E,K)>0$ such that ...
7
votes
1answer
627 views

Crystalline realizations of Artin motives

What are the crystalline realizations of Artin motives? In a paper by Kisin and Wortmann, "A note on Artin motives" (google it and you'll find it immediately), they define a suitable category of ...
15
votes
1answer
1k views

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 ...
5
votes
3answers
820 views

Crystalline Characters

Let $K$, $L$ be finite extensions of the $p$-adic numbers. Suppose $\chi:G_K\rightarrow L^{\times}$ is crystalline. It is my understanding that if either $K$ or $L=\mathbb{Q}_p$, then $\chi$ must be a ...
15
votes
2answers
2k views

$p$-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 ...
6
votes
1answer
694 views

Semisimple Weil-Deligne representations

I've just realized that I don't understand something important and basic about the Weil-Deligne group and its representations. (I'm not very surprised by this). Following Deligne's article, Section ...
10
votes
2answers
625 views

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 ...
7
votes
3answers
614 views

When is an extension of characters de Rham?

Let $G$ be the abolute Galois group of $\mathbb Q_p$, let $\delta_1, \delta_2: G\rightarrow L^{\times}$ be continuous characters, where $L$ is a finite extension of $\mathbb Q_p$. Assume that ...
12
votes
2answers
1k views

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. ...
11
votes
1answer
575 views

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) ...
12
votes
1answer
4k views

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)
10
votes
1answer
924 views

Tamely ramified p-adic Galois representations

The following question came up in a discussion with a colleague about local Galois representations: To what extent is the classification of continuous $p$-adic representations of ...
7
votes
1answer
490 views

On the determinant of an odd, continuous Galois representation.

In his paper, Duke paper, Serre consider continuous, odd Galois representation $\rho: G_{\mathbb{Q}}\longrightarrow GL_{n}(\overline{\mathbb{F}}_{p})$ where $p$ is a rational prime. Roughly, (I don't ...
5
votes
1answer
733 views

Generalization of Raynaud's (p, p, … p) result

Does Corollary 3.4.4 in Raynaud's paper ``Schemas en Groupes de Type (p, ..., p)'' apply also to the case where G is quasi-finite? If not, what is the more general statement? The corollary states: ...
10
votes
2answers
872 views

Automorphic forms and Galois representations over imaginary quadratic fields: generalizing Taylor's theorem?

Let $K/\mathbf{Q}$ be an imaginary quadratic field, with $\sigma \in \mathrm{Gal}(K/\mathbf{Q})$ a generator. Suppose $\pi$ is a cuspidal automorphic representation of $GL_2 / K$ with central ...
8
votes
0answers
289 views

Explicitly describing a two-dimensional reducible representation of G_{Q_p}

Let $j$ be some integer which is neither 0 nor 1. The Galois cohomology space $H^1({\mathbb Q}_p,{\mathbb Q}_p(j))$ is then 1-dimensional, and thus there is a unique non-split 2-dimensional local ...
20
votes
1answer
2k views

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 ...
4
votes
1answer
543 views

Serre's conjecture for mod-p^n representations?

I think this may be a silly question, but here goes. Let $\rho:\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to \mathrm{GL}_2(\overline{\mathbf{F}_p})$ be a representation; say $\rho$ is of S-type ...
10
votes
2answers
1k views

Are Kato's zeta elements integral?

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $T$ the $p$-adic Tate module of $E$. Kato's Euler system, constructed in the paper "P-adic Hodge theory and values of zeta functions of modular ...
7
votes
1answer
547 views

Can an etale (phi, Gamma) module be an extension of non-etale ones?

This question is about p-adic representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)$ and $(\varphi, \Gamma)$-modules. By theorems of Fontaine, Cherbonnier-Colmez and Kedlaya, the ...
11
votes
1answer
3k views

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 ...
6
votes
2answers
404 views

Integral models of p-adic representations

Let $G$ be a compact group and $K$ a finite extension of $Q_{p}$. If $\rho$ is a continuous representation of $G$ on a finite dimensional vector space over $K$, then it is well known that the ...
13
votes
0answers
781 views

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 ...
7
votes
1answer
365 views

Mod l local Galois representations (l different from p)

My question is referred to the statement and proof of Prop. 2.4 of Diamond's article "An extension of Wiles' Results", in Modular Forms and Fermat Last Theorem, page 479. More precisely: fix $l$ ...
22
votes
1answer
4k views

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 ...
4
votes
2answers
876 views

Galois theory of endomorphism rings of irreducible representations

Let $k$ be a field which I don't suppose to be algebraically closed. Then, the endomorphism ring of any irreducible representation of a finite group over $k$ is a division ring. What is known about ...
8
votes
2answers
950 views

Does the p-adic Tate module of an elliptic curve with ordinary reduction decompose?

Let $K$ be a finite extension of $\mathbb{Q}_p$ and $E$ an elliptic curve over $K$ with good ordinary reduction. The p-adic Tate module $T_p(E)$ is (after tensoring with $\mathbb{Q}_p$) a ...
1
vote
0answers
327 views

Ext of Tate-modules of abelian varieties

Let $K$ be a local field (in fact, finite extension of $\mathbb{Q}_p$) and let $A$ and $B$ be abelian varieties over $K$. Associated to $A$ and $B$ are the Tate-modules $T_p(A)$ and $T_p(B)$. Both ...