10 votes
Accepted

What is a semistable representation?

An E-rep of $G_K$ (E either $Q_p$ or $\bar{Q}_\ell$, K an extension of $Q_p$) being semistable basically means that it looks like a representation coming from the etale cohomology of a variety with ...
PL.'s user avatar
  • 1,906
7 votes
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The splitting of Galois representations

This is false for simple group theoretical reasons. Suppose that $V$ is an absolutely irreducible representation of a group $G$ which has odd dimension $d$ and which is self-dual up to twist, say $V \...
Furlo Roth's user avatar
7 votes
Accepted

Why does having an integral model make étale cohomology unramified?

This is really a local question: you may as well assume $X$ is defined over a local field $L$, it doesn't matter whether it comes from a number field. The point is that if $S = \operatorname{Spec} O_L$...
David Loeffler's user avatar
6 votes

The Frobenius Trace for an elliptic curve

The key fact you need is this: if $E/K$ is an elliptic curve which has complex multiplication over $K$, then the associated $\ell$-adic Galois representations $$\rho_{E,\ell}:G_K\to\mathrm{GL}_2(\...
Mathmo123's user avatar
  • 23k
6 votes

the corresponding modular form of an elliptic curve

Yes. (a) You compute the conductor of your elliptic curve (say using Tate's algorithm, or some variant). (b) You then compute all the weight two Hecke eigenforms at that level (using modular ...
tracing's user avatar
  • 5,510
6 votes

What are Robba rings and why are they important?

One sentence of buzzwords to help your literature search: it connects the field of norms (or, more fashionably, tilting) with Fontaine's functors of $p$-adic Hodge theory. The original work in this ...
user762732's user avatar
6 votes
Accepted

What is the Weil group of a global field $K$?

In general, the concept of a Weil group arises via the theory of class formations, with the standard introductory reference being Tate's Number theoretic background. In the case that the field is (...
P...'s user avatar
  • 141
6 votes
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Mod $p$-representation ($p$-torsion points ) of elliptic curve (over number fields) with CM can be irreducible?

Certainly. It will (typically) be irreducible for primes where the curve has supersingular reduction. Indeed, for such primes, the p-torsion as a module over the endomorphism ring $O$ is isomorphic to ...
iPhone user's user avatar
6 votes
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Why $\mathbb{Q}(x_1,y_1,...,x_n,y_n)/\mathbb{Q}$ is a Galois extension? where $E[m]=\{(x_1,y_1),...,(x_n,y_n)\}$ is the m-torsion group.

An algebraic field extension $L/K$ is Galois if it is normal and separable. In our case, we only need to show that $L/K$ is normal. However, normality is equivalent, that for every field homomorphism $...
Dietrich Burde's user avatar
5 votes
Accepted

irreduciblity of $\ell$-adic representation attach to the elliptic curve over $\mathbf{Q}$ with complex multiplication

An elliptic curve over $\mathbf Q$ cannot have complex multiplication (defined over $\mathbf Q$). It's possible for a rational elliptic curve to have extra endomorphisms, but these will only be ...
Mathmo123's user avatar
  • 23k
5 votes
Accepted

Reference to proof of Weil-Langlands theorem

As noted by Mathmo123 in comments, this is a form of the converse theorem. There is the original paper of Weil in which he proved this (but I'm not sure if he treated the weight 1 case which is ...
tracing's user avatar
  • 5,510
5 votes
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Definition of Galois representation unramified at a prime.

$\require{AMScd}$ If $\rho$ factors through some $\overline{\rho}\colon \mathrm{Gal}(L/K) \to \mathrm{GL}_n(F)$ with $L/K$ unramified at $\mathfrak{p}$, then the image of $I_{\mathfrak{p}}$ under $\...
Alessandro17's user avatar
5 votes
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topology on the ring of Witt vectors in the theory of period rings of Fontaine

There are (at least) two different topologies on all those rings, and that is the source of your confusion. Both on $W(R)$ and $W(fof(R))$ one has the $p$-adic topology, in which e.g. a basis of ...
Torsten Schoeneberg's user avatar
5 votes
Accepted

Property of a Galois representation associated to an elliptic curve

For an elliptic curve over $\Bbb{Q}$ or any field of characteristic $0$ $$x^3+ax+b = (x-e_1)(x-e_2)(x-e_3)$$ then since $-(x,y) = (x,-y)$ we see $(\infty,\infty),(e_1,0),(e_2,0),(e_3,0)$ is in the 2-...
reuns's user avatar
  • 78k
5 votes
Accepted

Fixed subfield of symmetric rational functions $K(s_1,\ldots,s_n)$ under $A_n$

You have the field $L$ with a faithful action of $S_n$. You have $F$, the fixed field of $S_n$ and $E$, the fixed field of $A_n$. Recall the theorem of Galois theory that states that if $G$ is a ...
Angina Seng's user avatar
4 votes
Accepted

Hilbert 90 and K-forms

Your presentation of the problem is a bit restricted (see below). As I understand it, a finite Galois extension $L/K$ and an $L$-algebra $A$ are given, $G=Gal(L/K)$ acts naturally on $Aut(A)$, and you ...
nguyen quang do's user avatar
4 votes
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Determinant representation of Tate module and cyclotomic character

I'm going to approach your questions in reverse order. For an abelian variety of dimension $g$, it is well known that the determinant representation of the Tate module is $\chi^g$, i.e. the $g$-th ...
Matt B's user avatar
  • 3,915
4 votes
Accepted

Integral models of $p$-divisible groups

Just to get this off the unanswered list. This is false even if you replace $\mathbb{C}_p$ by a finite extension of $\mathbb{Q}_p$. When I want to think of concrete examples of $p$-divisible groups ...
Alex Youcis's user avatar
  • 54.1k
4 votes
Accepted

Image of Galois representation of ordinary elliptic curve

This just boils down to the follow fact. Let $C \subset E[p]$ be a galois stable subgroup of order $p$, then the image of the representation $\rho$ is contained in a subgroup conjugate to the subgroup ...
Mummy the turkey's user avatar
4 votes
Accepted

Why do we restrict ourselves to continuous representations of galois groups?

This started as a comment, but it's grown a bit too long! For starters there are plenty of examples of discontinuous representations. For example, you'll often see number theorists fixing an ...
Mathmo123's user avatar
  • 23k
4 votes
Accepted

Is $H^1_{et}( \overline{C},\mathbb Q_l)$ irreducible as a Galois representation?

First of all, in complete generality, if $(\rho_\ell\colon G_{\mathbb Q}\to \mathrm{GL}_n(\overline{\mathbb Q}_\ell))$ is a compatible system of $\ell$-adic Galois representations, then, conjecturally,...
Mathmo123's user avatar
  • 23k
4 votes
Accepted

Method of associating Galois representation to normalized Hecke eigenform of weight 2 not work for other weights. Why?

Here's a slightly ahistorical take. The short answer is that we get lucky with weight $2$ modular forms, that their Galois representations are relatively easy to construct. As a general rule, ...
Mathmo123's user avatar
  • 23k
4 votes

Representations with traces in a finite field

Take $F=\mathbb{R}$ and the canonical representation of the quaternions $G\longrightarrow{\rm GL}_2(\mathbb{C})$ seen as a subgroup of ${\rm GL}_2(\mathbb{C})$. All the quaternion matrices have real ...
Tuvasbien's user avatar
  • 8,877
3 votes

Characterization for the continuity of Weil representations

Let's take a step back for a second and ask: if $H$ is a discrete topological group and $G$ is an arbitrary topological group, what does it mean for a homomorphism $\rho:G\to H$ to be continuous? ...
Mathmo123's user avatar
  • 23k
3 votes
Accepted

Galois representation associated with p-ordinary modular form.

For any $v\in V$, the orbit map $g\mapsto gv:G_{\mathbf{Q}}\to V$ is continuous; if we follow this with the continuous surjection $V\to V/T$, we infer that the orbit map $g\mapsto g(v+T):G_{\mathbf{Q}}...
Keenan Kidwell's user avatar
3 votes
Accepted

Freeness of Tate module of the multiplicative group

$\Bbb Z_\ell\cong\varprojlim \Bbb Z/\ell^n\Bbb Z$, where the map $\Bbb Z/\ell^{n+1}\Bbb Z\to\Bbb Z/\ell^n\Bbb Z$ is the natural map taking $n\pmod{\ell^{n+1}}$ to $n\pmod{\ell^n}$ (maybe by definition,...
Stahl's user avatar
  • 23.2k
3 votes

Why is the kernel of a Galois representation an open subgroup?

Suppose that $\ker(\rho)$ is open. If $U$ is an open subset of $\mathrm{Aut}_E(V)$ and $g\in \rho^{-1}(U)$, then $g\ker(\rho)$ is an open subset of $\rho^{-1}(U)$. Hence, $\rho^{-1}(U)$ is open, so $\...
Mathmo123's user avatar
  • 23k
3 votes

Why is the image of a $\pmod p$ Galois representation finite?

The algebraic closure $\overline{\mathbb{F}_q} = \overline{\mathbb{F}_p}$ is naturally the direct limit of all the intermediate extensions: $$ \overline{\mathbb{F}_p} = \varinjlim_n \mathbb{F}_{p^n} $$...
Alessandro17's user avatar
3 votes

Importance of continuity of Galois representations

The comment by @Mariano Suárez-Alvarez "By definition, continuity is equivalent to factoring through a finite quotient." is completely wrong. In particular, there are uncountably many surjective ...
user335783's user avatar

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