# Tagged Questions

5answers
185 views

### Applications of Character Theory

Some of the applications of character theory are the proofs of Burnside $p^aq^b$ theorem, , Frobenius theorem and factorization of the group determinant (the problem which led Frobenius to character ...
0answers
73 views

### Representations of $\text{GL}_2(\mathbb{Q})$

Let's say that as a representation theorist I am naively interested in representations of $G(\mathbb{Q})$, where $G$ is an algebraic group defined over $\mathbb{Q}$. For the purposes of this question, ...
1answer
46 views

### Some questions about representation theory in the modular case

I'm working on a paper which uses representation theory in order to compute some characters and deduce arithmetical statements about certain field extensions. Let $\Delta$ be a group of order prime ...
1answer
277 views

### Undergraduate roadmap for Langlands program and its geometric counterpart

What are the topics which an undergraduate with knowledge of algebra, galois theory and analysis learn to understand Langlands program and its goemetric counterpart? I would also like to know what are ...
0answers
57 views

### A natural way of thinking of the definition of an Artin $L$-function?

Emil Artin knew that given a finite extension of $L/\mathbb{Q}$, the local factor of the zeta function $\zeta_{L/\mathbb{Q}}$ at the prime $p$ should be \$\displaystyle\prod_{\mathfrak{p}|p}\frac{1}{1 ...
2answers
115 views

### Property (T) and Number Theory?

Pardon the wild question, but Are there any known connections between Kazhdan's property (T) and number theory, or number theoretic consequences of property (T)? Edit: To clarify, I'd like to ...
1answer
254 views

### What is the idea of a monodromy?

Is there a connexion between : 1) The monodromy group of a topological space. 2) The $\ell$-adic monodromy theorem of Grothendieck. 3) The $p$-adic monodromy conjecture of Fontaine (which is now ...
1answer
66 views

### Why does rigidity hold only if rank >1?

In simple words, why does Margulis' superrigidity and arithemiticit only hold for lattices in Lie groups of rank $>1$? E.g. what is the reason for it to fail for $SL(2,R)$?
1answer
166 views

### Does the $p$-torsion of an elliptic curve with good reduction over a local field always determine whether the reduction is ordinary or supersingular?

Let $K$ be a finite extension of $\mathbb{Q}_p$ and $E/K$ an elliptic curve with good reduction. Does the $\mathbb{F}_p[\mathrm{Gal}(\overline{K})]$-module $E[p](\overline{K})$ determine whether the ...
1answer
166 views

### 2-dimensional $\ell$-adic representations [closed]

In an assignment, I have to give an example of a 2-dimensional $\ell$-adic representation of the absolute Galois group of $\mathbb{Q}$, bu I am faced with the problem that I do not a lot of these. Or ...
2answers
230 views

### What is the standard definition of an ordinary (local) $p$-adic Galois representation?

Let $V$ be a $n$-dimensional $\mathbf{Q}_p$-vector space with a continuous action of $\operatorname{Gal}(\bar{L}/L)$, where $L$ is a complete discretely valued field of characteristic zero with ...
1answer
75 views

### Where can I find rigorous statements about the spectral decomposition of reductive groups?

Given a global field $F$ and a reductive group $G$, where can I find the spectral decomposition of $$L^2( Z(\mathbb{A}) G(F) \backslash G( \mathbb{A})).$$ I will need the result in this generality, ...
2answers
66 views

### Eisenstein spectrumfor $GL(n)$

Fix a global field $F$. Does every automorphic representation of $GL(n)$ appear as an arbitrary twists in the continuous spectrum of $GL(m)$, $m>n$? What happens for the automorphic ...
3answers
248 views

### Complex Galois Representations are Finite

In A First Course in Modular Forms, Diamond and Shurman leave as an exercise ($9.3.3$) that every complex Galois representation is finite. While I think I have worked through this exercise here, this ...
1answer
214 views

### What is a description for the following number theoretic object?

The title couldn't quite contain the question, so I didn't attempt to make it precise. I should note that this is the third or fourth question I've asked these past two days about problems I've been ...
1answer
132 views

### Questions about p-adic representations

In a paper I'm currently reading, they have the following situation: $k$ is some number field that doesn't have a primitive $p^{th}$ root of unity, and $k(\zeta_p)$ a field above it with Galois group ...
1answer
230 views

### torsion representation

Let $\mathbb{Z}_p$ be te ring of p-adic integers and let $T$,$T'$ be two free $\mathbb{Z}_p$-module with a continuous action of $G_{\mathbb{Q}_p}$ (the absolute Galois group of $\mathbb{Q}_p$). It is ...
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