# Tagged Questions

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### Isometric representation semisimple?

The first lemma on p.35 of these notes states that unitary representations are semisimple. Could the same be said of isometries if the space doesn't have an inner product? This topic notes that the ...

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**1**answer

322 views

### Explicit calculation of Weil Deligne representations

According to Grothendieck monodromy theorem, l-adic galois representations of a local field corresponds to Weil-Deligne representations.
However, given a galois representation, it is usually difficult ...

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306 views

### Irreducible representations of compact groups

Let G be a compact group (or even profinite - Galois group). Let $V$ be a vector space over the field ${\mathbb F}_p$ with $p$ elements, $p$ a finite prime, such that $V$ is a contable product of ...

**5**

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**1**answer

135 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 ...

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214 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 ...

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82 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 $P$, ...

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175 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 ...

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**1**answer

1k 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 ...

**3**

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**1**answer

212 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 ...

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**3**answers

999 views

### Are there Maass forms where the expected Galois representation is $\ell$-adic?

Recall that by theorems of Deligne and Deligne--Serre, there is the following dichotomy:
Modular forms on the upper half plane of level $N$ and weight $k\geq 2$ correspond to representations ...

**10**

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**2**answers

978 views

### “Purely local” proof of local Langlands

As from this website
http://math.uchicago.edu/~lxiao/workshop_site/
My question is: What does it mean by "purely local"?
Also, I heard about this phrase "purely local" in other problems as well, ...

**2**

votes

**1**answer

291 views

### $l$-adic representations from Shimura curves

This question may be kind of vague. And we use the same notations as in Carayol's papers:
H. Carayol, Sur les représentations l-adiques associées aux formes modulaires de Hilbert;
H. Carayol, Sur la ...

**20**

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**3**answers

1k views

### Subgroups of GL(2,q)

I have been wondering about something for a while now, and the simplest incarnation of it is the following question:
Find a finite group that is not a subgroup of any $GL_2(q)$.
Here, $GL_2(q)$ ...

**1**

vote

**1**answer

212 views

### Dimension of fixed points of Galois group actions

I have a question about fixed points of Galois group actions.
I am hoping that this is easy for the experts.
Let $k$ be a field of characteristic $0$. Let $K$ be a finite
Galois extension of $k$ ...

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300 views

### Which properties of p-adic representations can be recovered from (\varphi,\Gamma)-modules?

In Berger's "An introduction to the Theory of $p$-adic Representations", he mentions that due the the equivalence of categories between etale $(\varphi,\Gamma)$-modules and $p$-adic representations, ...

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**1**answer

530 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 ...

**4**

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**2**answers

817 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**

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**2**answers

661 views

### number of irreducible representations over general fields

For a finite group, there are finitely many irreducible representations of complex numbers.
What if the field is changed to some other fields? Like real numbers, p-adic field, finite field?
In ...