The galois-representations tag has no wiki summary.

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### 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 ...

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**1**answer

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### 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 ...

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### 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? ...

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328 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 ...

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499 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 ...

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votes

**1**answer

362 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 ...

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200 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 ...

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565 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
...

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613 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 ...

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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 ...

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765 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 ...

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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 ...

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670 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 ...

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593 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 ...

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580 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 ...

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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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566 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) ...

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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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### 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 ...

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473 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 ...

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701 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:
...

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856 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 ...

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### 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 ...

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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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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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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### 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 ...

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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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### 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 $l$ ...

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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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### 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 ...

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### 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 ...

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314 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 ...

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### When do the Galois reps of modular forms have open image?

Suppose f is a newform (with coefficients generating some number field E), and $\rho_{f,\lambda}: {\rm Gal}(\overline{\mathbb{Q}} / \mathbb{Q}) \to {\rm GL}_2(E_\lambda)$ the associated Galois rep ...

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### Images of action of Galois on the Tate module of Elliptic Curve,

Let E be an elliptic curve over the rationals, and let $TE = \lim_\leftarrow E[n]$ be the Tate module of the elliptic curve. The action of the Galois group of $\bf Q$ gives rise to a representation ...

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### How canonical are localization maps in Galois cohomology?

The setup for my question is as follows: $k$ is a field, $K$ a Galois extension of $k$ with group $G$, $k^\prime$ an arbitrary extension of $k$, and $K^\prime/k^\prime$ another Galois extension of ...

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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. ...

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### P-adic local Langlands for non-unitary representations?

In Colmez's work on the p-adic local Langlands correspondence for ${\rm GL}_2(\mathbb{Q}_p)$, he works with ${\rm GL}_2(\mathbb{Q}_p)$-representations on $p$-adic Banach spaces which admit an ...

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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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552 views

### What are the maximal subgroups of GSp(2g,l)?

Is there a nice description of the maximal subgroups of $GSp_{2g}(\mathbb{F}_l)$?
When $g = 1$ this is $GL_2(\mathbb{F}_l)$, and Serre (in his 72 Inventiones paper) classifies its maximal subgroups ...

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### What is the image of complex conjugation under Siegel Galois representations?

Let $G$ be the reductive group $\operatorname{GSp}_{4}$. Let $\pi$ be a smooth admissible cuspidal representation of $\operatorname{GSp}_{4}(\mathbb{A}^{(\infty)})$ of dominant weight. Assume, for ...

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### Is there an R=T type result for modular forms with additive reduction?

Let E be an elliptic curve over the rationals with conductor $Mp^2$ with p>5 and M and p coprime, and let $\rho$ be the Galois representation attached to the p-torsion points of E. Is there a way to ...

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### In what sense (if any) does the cohomology of profinite groups commute with projective limits?

Background:
Let $G$ be a profinite group. If $M$ is a discrete $G$-module, then $M=\varinjlim_U M^U$, where the direct limit is taken with respect to inclusions over all open normal subgroups of $G$, ...

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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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### What geometric properties do properties of ell-adic Galois representations imply?

This is the converse question to an earlier question. More precisely,
Let $X/K$ be a curve(or variety) over a global field $K$. We consider the Galois representation obtained by the absolute Galois ...

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### When is the Galois representation on the étale cohomology unramified/Hodge-Tate/de Rham/crystalline/semistable?

Let $X/K$ be a variety over a global field $K$. When (and why) is the Galois representation $H^i_{et}(X \times_K \bar{K}, \mathbf{Q}_\ell)$ unramified at a place $v$ of $K$?
I guess this is true if ...

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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 ...

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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) ...