**5**

votes

**1**answer

574 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

**2**answers

926 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

**1**answer

801 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

**0**answers

497 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

**1**answer

407 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

**1**answer

823 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

**1**answer

915 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

**1**answer

744 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

**3**answers

814 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? ...

**3**

votes

**2**answers

340 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

**1**answer

555 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

**1**answer

381 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

**0**answers

216 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

**0**answers

679 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

**1**answer

660 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 ...

**16**

votes

**1**answer

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 ...

**6**

votes

**3**answers

960 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 ...

**16**

votes

**2**answers

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

**1**answer

723 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 ...

**11**

votes

**2**answers

751 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

**3**answers

654 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 ...

**13**

votes

**2**answers

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

**1**answer

591 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

**1**answer

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

**1**answer

1k 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 ...

**8**

votes

**1**answer

552 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

**1**answer

790 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

**2**answers

909 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

**0**answers

303 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 ...

**23**

votes

**1**answer

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

**1**answer

558 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

**2**answers

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

**1**answer

621 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 ...

**13**

votes

**1**answer

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

**2**answers

410 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 ...

**16**

votes

**0**answers

897 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

**1**answer

387 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 $l$ ...

**24**

votes

**1**answer

5k 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

**2**answers

971 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 ...

**10**

votes

**2**answers

1k 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

**0**answers

351 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 ...

**8**

votes

**2**answers

673 views

### 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 ...

**8**

votes

**1**answer

700 views

### 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 ...

**7**

votes

**1**answer

393 views

### 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 ...

**15**

votes

**1**answer

1k views

### 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. ...

**9**

votes

**1**answer

1k views

### 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 ...

**11**

votes

**1**answer

722 views

### 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 $O_K$, ...

**7**

votes

**4**answers

575 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 ...

**3**

votes

**1**answer

457 views

### 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 ...

**7**

votes

**1**answer

394 views

### 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 ...