8
votes
0answers
195 views

Irreducibility of Galois representations attached to unitary groups

If $G$ is a unitary group in $n$ variables over $\mathbb Q$, attached to an hermitian form for an imaginary quadratic extension $E/\mathbb Q$ and if we suppose that the hermitian form is definite over ...
8
votes
0answers
107 views

Algebraic construction of the modular representation of $\mathrm{SL}_2(\hat{\mathbf Z})$

The answer to this question is probably to be found in the theory of automorphic forms, but (I don't know much about it and consequently) after some tries, I did not catch it. Thus I'd be grateful if ...
7
votes
0answers
141 views

Phillips-Sarnak conjecture in higher dimension

The Phillips-Sarnak conjecture states that for a generic Fuchsian lattice the space of Maass cusp forms is finite-dimensional. Generic here means in particular non-uniform, non-arithmetic, no special ...
5
votes
1answer
81 views

Name or references for minimal $N$ such that $\left(\frac{a}{b}\right)_n = \left(\frac{a}{b'}\right)_n$ whenever $b \equiv b' \bmod (N)$

Let $\left( \dfrac{a}{b} \right)_n$ denote the nth power residue symbol, a generalization of the Legendre symbol. I have recently seen it quoted that there is a minimal ideal $N$ (minimal by ideal ...
4
votes
1answer
274 views

To which automorphic forms/rep's over a function field can we associate a Galois representation?

As far as I understand it, by the work of Lafforgue (cf. Laumon, "Cohom. of Drinfeld ... II", Thm 12.4.1) there is a Galois representation associated to an irreducible cuspidal automorphic ...
1
vote
6answers
925 views

List of structure theorems for vector valued Siegel modular forms (esp. of genus 2)

What are Siegel modular forms? We start with defining their common domains $\mathbb{H}_g$ as the set of symmetric $g \times g$ matrices with positive definite imaginary parts. The symplectic group ...
9
votes
1answer
376 views

Borel's Paris Lectures

I am trying to read Harish-Chandra's book on automorphic forms on Semisimple Lie groups, and he keeps referring to Borel's Paris lecture notes. Does anyone have an online version of these notes or ...
3
votes
1answer
221 views

Volume of PGL(2,F) \ PGL(2, A)

Let $F$ be a global field. What is the measure of $PGL_2(F) \backslash PGL_2(\mathbb{A})$? This depends of course on the normalizations of the Haar measures on $PGL_2(F)$ and $PGL_2(\mathbb{A})$. ...
6
votes
1answer
291 views

What is the support of the Whittaker function of a new vector on GL(2)?

Let $W$ be the normalized Whittaker function associated to a new vector in an irreducible generic representation $\pi$ of $G=GL_2(k)$, where $k$ is a $p$-adic field. Let $c$ be the conductor of $\pi$, ...
4
votes
1answer
180 views

Half integral weight Hecke operators

I would like to find a source giving the exact formula for the product of two Hecke operators $T_{\kappa}(n^2)$ and $T_\kappa(m^2)$ of half integral weight. That is, $\kappa \in \frac 12 \mathbb{Z} - ...
2
votes
1answer
235 views

Pseudo coefficients and orbital integrals

I am looking for a reference/idea, how this passage from Labesse's Snowbird Lecture "Introduction to endoscopy" pg.5 can be explained: "We shall denote by $f_\pi$ a pseudo-coefficient for $\pi$, ...
12
votes
3answers
1k views

Questions about the Bernstein center of a $p$-adic reductive group

Dear all, The "Bernstein center" of a $p$-adic reductive group appears frequently in the literature of automorphic forms, often without a precise definition. For example, in page 233 of ...
11
votes
4answers
394 views

Growth of smallest closed geodesic in congruence subgroups?

Let $\Gamma$ be one of the classical congruence subgroups $\Gamma_0(N)$, $\Gamma_1(N)$ and $\Gamma(N)$ of $SL(2, \mathbb{Z})$. How does the lower bound for the length of primitive geodesics on ...
7
votes
2answers
1k views

What is the non-motivic motivation behind automorphic representations?

In one of my last questions: What is the "reason" for modularity results? it was pointed out to me that "the notion of automorphic representation developed independently of any concern with ...
28
votes
6answers
4k views

How is representation theory used in modular/automorphic forms?

There is certainly an abundance of advanced books on Galois representations and automorphic forms. What I'm wondering is more simple: What is the basic connection between modular forms and ...
4
votes
1answer
608 views

Rallis inner product formula for U(2,2) and U(3)

Victor Tan has a couple of papers on a regularized Siegel-Weil formula for U(2,2) and U(3). The papers I'm talking about are: "A Regularized Siegel-Weil Formula on U(2,2) and U(3)", Duke, 1998. "An ...