8
votes
0answers
140 views

Symmetric Fifth Power Lift of GL(2) Automorphic Form

Let $\pi$ be an automorphic representation of $GL(2)/\mathbb{Q}$. For simplicity, you can take it to be a Maass form for $SL(2,\mathbb Z)$. Kim, Shahidi, Gelbart-Jacquet prove that $$L(s, \pi, ...
8
votes
1answer
169 views

Self-dual automorphic forms on GL(4)

as is known among experts, all self-dual automorphic form on GL(3) comes from symmetric square lift from GL(2). You can find this from Ramakrishnan ...
4
votes
1answer
170 views

looking for reference on dihedral, tetrahedral, or octahedral forms

I am looking for a reference on dihedral, tetrahedral, or octahedral forms. As far as I read, they are some cuspidal automorphic forms on $GL(2)$ induced from $GL(1)$. Dihedral is from $GL(1)/K$ to ...
7
votes
1answer
237 views

Is the adjoint L-function on GL(m) holomorphic?

Let $\pi$ be an automorphic representation on GL($m$)/$\mathbb{Q}$. Define $$L(s,\pi,Ad):=\frac{L(s,\pi\times\overline{\pi})}{\zeta(s)}.$$ This is an L-function with euler products of degree $m^2-1$. ...
5
votes
1answer
204 views

absolute convergence of Rankin-Selberg series

Let $\pi$ and $\pi'$ be two general automorphic representation on $GL(n)$ and $GL(n')$ over $\mathbb{Q}$. I heard that the rankin-selberg convolution L-function $L(s,\pi\times\pi')$ is absolutely ...
10
votes
2answers
335 views

Langlands' original observation about Ramanujan conjecture

Obviously functoriality of arbitrary high symmetric power lifts of automorphic forms on GL(2) will lead to the Ramanujan conjecture. But I guess that is too strong for Ramanujan. I came across some ...
7
votes
1answer
170 views

standard zero free region of automorphic L-function on GL(N)

Let $L(s,\pi)$ be the standard(Godement-Jacquet) $L$-function of $\pi$, where $\pi$ is a cuspidal automorphic represetation of $GL(m,A_Q)$. What's the standard zero-free region for $L(s,\pi)$? any ...
0
votes
1answer
149 views

Euler product of Asai L-function?

Let $\pi$ be an automorphic form of GL(n)/$\mathbb{Q}$ with standard $L$-function $$L(s,\pi)=\prod_p \prod_{i=1}^n(1-\frac{\alpha_{p,i}}{p^s})^{-1},$$ where $\{\alpha_{p,i}:i=1,\dots,n\}$ are the ...
3
votes
0answers
76 views

Functoriality for triple product GL(2) x GL(2) x GL(2)

Let $f$, $g$ and $h$ be three general automorphic forms on GL(2). Do we know that $L(s, f\times g\times h)$ comes from an automorphic form on GL(8)?
9
votes
0answers
311 views

What's the status of Arthur's announced classification for GSp(4)?

In "Automorphic representations of GSp(4)" (2004) (see http://www.math.toronto.edu/arthur/), James Arthur announces a classification of discrete automorphic representations of GSp(4). There are no ...
4
votes
0answers
181 views

Why Whittaker functions are useful?

Whittaker functions appears in Langlands program. Recently, it is shown that some Whittaker functions can be obtained by integrating a function related to decoration over a geometric crystal in ...
8
votes
0answers
293 views

The status of automorphic induction

Background: Let $K/F$ be a degree $r$ extension of number fields. It is conjectured that an automorphic representation of GL$_n$ associated to $K$ induces an automorphic representation of GL$_{rn}$ ...
8
votes
1answer
519 views

best record toward Selberg's eigenvalue conjecture?

What's the best record toward Selberg's eigenvalue conjecture: Maass forms on $\Gamma_o(N)$ has eigenvalue greater than 1/4?
4
votes
0answers
103 views

parametrization of irreducible finite dimensional representation of Weil group

Let $F$ be a p-adic field, with p a prime denoting the residue field characteristic. Let $\mathcal{W}_F$ be the Weil group. In the local Langlands correspondence for $GL(n,F)$, it is important to know ...
13
votes
1answer
659 views

What is the Twisted Trace Formula?

I am studying the trace formula using "An Introduction to the Trace Formula" by James Arthur. I would like to understand the twisted trace formula, but unfortunately I never came across a good ...
3
votes
0answers
99 views

Do local L-functions/epsilon factors vary continuously with the Fell topology?

Edit due to the comment. Consider $G=GL(2)$ over a local field $F$. The Fell topology on the unitary dual of $G(F)$ is seperable. Given a sequence of irreducible unitary representations $(\pi_n)$ of ...
5
votes
0answers
359 views

a generalization of a formula of Shimura

Let $\phi$ be a $GL(2)$ automorphic form with Fourier coefficients $a(n)$ and $a(1)=1$. Obviously we have $L(s,\phi)=\sum \frac{a(n)}{n^s}$. Shimura have the following formula $L(s, Ad\; ...
2
votes
0answers
128 views

Conceptual reason behind Shimura lifts

Shimura lifts are correspondence between integer weight and half-integral weight automorphic forms. Half integral weight things are associated to representation of a double cover of $G ...
0
votes
0answers
295 views

Local Langlands conjecture for GL(2)

Let $F_v$ be a local field. Let $\sigma_v$ be a two-dimensional representation of $Gal(\overline{F}_v:F_v) \rtimes W_{F_v}$. Now, there exists an infinite-dimensional representation $\pi_v$ of ...
10
votes
2answers
1k views

New Geometric Methods in Number Theory and Automorphic Forms

The MSRI is organising a programme with the above title from Aug 11, 2014 to Dec 12, 2014. Here is a short description from their website : The branches of number theory most directly related ...
1
vote
1answer
94 views

Maass-Hecke construction

I heard this name that it can construct GL(2) automorphic forms or L-functions from GL(1)? I did not find it anywhere. Or does it have another name which we are familiar with?
9
votes
1answer
398 views

Request for errata for Automorphic Forms on GL(2)

Edit (7/21/2014): We have finished proofreading Jacquet-Langlands and posted it to Robert Langlands's publications site. If you would like a copy, please download it from here: ...
3
votes
0answers
183 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 ...
15
votes
1answer
980 views

Understanding the “idea” behind Langlands

Apologies in advance if this is a bit too simple to ask here, but I think I'm probably more likely to get an answer here than at stackexchange. I've been trying to learn the basics of the Langlands ...
2
votes
0answers
239 views

on the fundamental lemma

I consider the fundamental lemma for the spherical Hecke algebra. Let $G$ a connected reductive quasisplit group on $F$, a local field of equal characteristic $p$. and $H$ an endoscopic group. Can ...
0
votes
1answer
218 views

Reference on Casselman-Shalika formula for GL(n) and PGL(n)?

I am looking for reference on Casselman-Shalika formula for GL(n) and PGL(n) at finite place p.
10
votes
0answers
1k views

Why doesn't functoriality immediately imply the modularity theorem?

Let $E/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, the prime indexed coefficients of its $L$-function agree with those of a weight $2$ cusp eigenform $f$ with integer coefficients. ...
2
votes
1answer
420 views

When is compact induction in GL(2) from an open compact group admissible?

Let $G$ be a locally profinite group and $K$ an open compact subgroup (mod the center), then Bushnell has shown that the following three statements are equivalent for a finite dimensional ...
2
votes
3answers
862 views

Automorphic Forms on product of groups $G\times H$

Dear all, I have some difficulty in understanding the notion of automorphic forms on product of groups. Let $G$, $H$ be two reductive groups defined over a number field $F$. Let $\mathcal{A}(G)$ be ...
4
votes
3answers
750 views

The historical development of automorphic geometry

Background: Today the notion of automorphic geometry is often framed in the context of the Langlands program, in particular what is sometimes called the Langlands reciprocity conjecture. This is ...
2
votes
2answers
569 views

Elliptic orbital integral

Let $F$ be a local field and $G = GL(n,F)$. Let $f$ be an element $C_c^\infty(G)$. Let $\gamma$ be an elliptic element of $G$ with irreducible characteristic polynomial. What are strategies to ...
8
votes
1answer
800 views

Is this a subcase of the fundamental lemma?

Let $F$ be a local field and $G= GL(n,F)$. Assume that $\gamma$ is an element of $G$ and $G_\gamma$ is its centralizer. The orbital integral is defined as $$ O_\gamma^G( \phi) = ...
2
votes
1answer
476 views

Different cuspidal automorphic representations with same representations at infinity

Let us fix a representation $\pi_\infty$ of GL(n,$\mathbb R$). Let us fix a character $\chi$ of K, where K is a compact subgroup of $GL(n,\mathbb A_{finite})$. $$K=\Pi_{v<\infty}K_v$$ $K_v$ is ...
3
votes
1answer
442 views

classification of irreducible admissible (g,K)-module for GL(3,R)

classification of irreducible admissible (g,K)-module for GL(3,R) Is there a classification of irreducible admissible (g,K)-module for GL(3,R)? For GL(2,R) we have principal series, discrete series ...
16
votes
4answers
1k views

How badly can strong multiplicity one fail in the theory of automorphic representations?

Let $G$ be a connected reductive group over a global field $k$, and let $\pi=\otimes_w\pi_w$ and $\pi'=\otimes_w\pi'_w$ be two automorphic representations for $G$, where here of course $w$ is ranging ...
4
votes
2answers
504 views

What is the relation of the Kuznetsov-Bruggeman trace formula and the Selberg trace formula?

I have read that there is an elementary way to show that the above mentioned trace fromulas are equivalent in the sense, that each of them can be derived directly from the other. There should exist a ...
3
votes
1answer
873 views

Jacquet Langlands correspondance

I have one issue with the Jacquet Langlands correspondance. The Weyl law for $H$ modulo a congruence subgroup and the Weyl law for cocompact groups are different. So why does this not contradict this ...
4
votes
1answer
1k views

Why is the Arthur trace formula so powerful?

Considering the Arthur trace formula, why are the sort of convolution operators, whose "normalized traces" are given in geometric terms and spectral terms, actually able to distinguish all automorphic ...
4
votes
2answers
614 views

Why is the simple trace formula a weaker tool than the Arthur trace formula?

What are some concrete examples of theorems which can be deduced from the Arthur trace formula, which do not follow from the simple trace of Kazdhan and Flicker? (So I do not mean weaker in the sense ...
35
votes
2answers
2k views

Langlands in dimension 2: the Yoshida conjecture

Background: One prominent part of the Langlands program is the conjecture that all motives are automorphic. It is of interest to consider special cases that are more precise, if less sweeping. ...
8
votes
0answers
469 views

Automorphic representations attached to abelian varieties

Let $A$ be an abelian variety defined over $\mathbb{Q}$, of dimension $d$. It is widely expected that there is an automorphic representation $\pi_A$ of $GL(2d)/\mathbb{Q}$ whose L-function agrees ...
10
votes
1answer
774 views

Simple explicit example of local Jacquet-Langlands theorem for inner forms of GL(n), and consequences

This one will be very easy for the experts. Let $F$ be a nonarch local field, let $n\geq1$ be an integer, choose $0\leq d<n$ and let $D$ be the central simple algebra over $F$ with invariant $d/n$ ...
41
votes
6answers
3k views

What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$-adic field?

Let me stress that I am only interested in $p$-adic fields in this question, for reasons that will become clear later. Let me also stress that in some sense I am basically assuming that the reader ...
22
votes
5answers
4k views

Where stands functoriality in 2009?

Robert Langlands is famous in number theory for making famous and deep conjectures about very abstract things called automorphic forms, somewhere in the 60s. There's a very interesting article by ...