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42
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 ...
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. ...
30
votes
3answers
2k views

Are there motives which do not, or should not, show up in the cohomology of any Shimura variety?

Let $F$ be a real quadratic field and let $E/F$ be an elliptic curve with conductor 1 (i.e. with good reduction everywhere; these things can and do exist) (perhaps also I should assume E has no CM, ...
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 ...
27
votes
3answers
2k views

What is the difference between an automorphic form and a modular form?

This is more of a question about terminology than about math. The term "automorphic form" is clearly a generalization of the term "modular form." What is not clear is exactly which generalization it ...
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 ...
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 ...
15
votes
3answers
1k views

Non-vanishing of p-adic L-functions

In Non-vanishing of L-series of modular forms (easy case?) it was answered that for a cuspidal newform $f$ of weight strictly greater than 2, then $L(f,1)$ is non-zero. (Here the $L$-series is ...
15
votes
1answer
992 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 ...
14
votes
3answers
1k 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 ...
14
votes
1answer
465 views

Is a reductive adelic group a Type I group?

I foresee that to experts of automorphic forms this question will sound unimportant or useless or even not worthy of an answer; but none of these are going to stop me from asking it! The question is ...
14
votes
2answers
1k views

Overview of automorphic representations for $SL(2)/{\mathbf{Q}}$?

In short: what does Labesse-Langlands say? Slightly more precise: what are the cuspidal automorphic representations of $SL_2(\mathbf{A}_{\mathbf{Q}})$, together with multiplicities? Let's say that I ...
13
votes
2answers
769 views

Why isn't meromorphic continuation enough for converse theorems?

This is a very naive question which really does little more than highlight my ignorance of how converse theorems really work. Take an algebraic gadget which should be conjecturally associated to an ...
13
votes
3answers
1k views

Constructing coherent sheaves on Shimura varieties.

Let me first run through the setting of my question in an example I understand well; that of modular curves. If $Y_1(N)$ denotes the usual modular curve over the complexes, the quotient of the upper ...
13
votes
3answers
1k views

Nonvanishing of central L-values of quadratic twists?

Let $\pi$ be a cuspidal automorphic representation of GL(2) over a number field (if you want, assume it's $\mathbb Q$ and $\pi$ comes from a holomorphic modular form). In the case $\pi$ has trivial ...
13
votes
3answers
579 views

Why only half-integral weight automorphic forms?

Why is that the theory of automorphic forms concentrates on the case of half-integral weight? I read in Borel's book "Automorphic forms on $SL_2$" (Section 18.5) that by considering the finite or ...
13
votes
4answers
1k views

Meromorphic continuation of Eisenstein series

I am interested what kind of (different) proofs of meromorphic continuation Eisenstein series (for general parabolic subgroups) exist in the literature? The only one I understand well, is Bernstein's ...
13
votes
1answer
670 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 ...
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 ...
12
votes
3answers
659 views

There is no lattice in PSL(2,R) which contains PSL(2,Z) properly?

How can I see that there is no lattice in $G=\mathrm{PSL}_2( \mathbb{R})$ which contains $\Gamma_1=\mathrm{PSL}_2( \mathbb{Z})$ properly, or equivalently, that $X_1 =\mathrm{PSL}_2(\mathbb{Z}) ...
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 ...
11
votes
4answers
1k views

Where do the real analytic Eisenstein series live?

In obtaining the spectral decomposition of $L^2(\Gamma \backslash G)$ where $G=SL_2(\mathbb{R})$, and $\Gamma$ is an arithmetic subgroup (I am satisfied with $\Gamma = SL (2,\mathbb{Z})$) we have a ...
11
votes
1answer
260 views

Hecke-module structure implicit in definition of automorphic forms in Borel-Jacquet's Corvallis article

Let $G$ be a connected reductive group over a number field $F$, $G_\infty=\prod_{v\mid\infty} G(F_v)$, $\mathbf{A}$ the adèles of $F$, $\mathbf{A}_f$ the finite adèles of $F$. Fix a maximal compact ...
11
votes
1answer
236 views

What can the theory of automorphic forms for $SL(n,\mathbb{Z})$ say about $SL(n,\mathbb{Z})$?

While reading "Automorphic Forms and L-functions for the Group $GL(n,R)$" by D. Goldfeld, I've got a feeling that linear groups over $\mathbb{R}$ and $\mathbb{Z}$ are considered only as technical ...
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 ...
10
votes
2answers
695 views

modular form Fourier coefficients and associated automorphic representation

Hi, Let $f$ be a cuspidal modular form of some weight and level $N$. Then it determines an irreducible automorphic representation $\pi = \bigotimes'\pi_p$ of $GL_2(\mathbf Q)$. Let $f = \sum_i a_i ...
10
votes
1answer
1k views

Which Shimura varieties are known to be automorphic?

This seems like something that should be well-known, but as an outsider to the field, I'm having trouble locating precise statements. Hasse-Weil zeta functions of Shimura varieties should be ...
10
votes
2answers
345 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 ...
10
votes
2answers
940 views

Most understandable notes on Jacquet-Langlands?

I am particularly interested in the comparison of the trace formula part of Jacquet-Langlands. But I found the original text hard to read.
10
votes
1answer
552 views

Double coset spaces of reductive groups and integral representations of L-functions

Let $G$ be a reductive group over a number field $k$, with center $Z$. Let $P$ be a parabolic subgroup. Let $H$ be a reductive subgroup of $G$. To what extent can we understand the double coset space ...
10
votes
1answer
326 views

Is the universal elliptic curve $\overline M_{1,2}$ a toric stack?

It is well-known that the compactification $\overline M_{1,1}$ of the moduli space of elliptic curves over $\mathbb C$ is a weighted projective line with weights $4$ and $6$. As far as I can tell, ...
10
votes
1answer
779 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 $d/n$ ...
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. ...
9
votes
2answers
512 views

Is there a canonical notion of “mod-l automorphic representation”?

As the title says. In particular, I am interested in the story for a general reductive group $G$, say defined over $\mathbb{Q}$. I can imagine that mod-$\ell$ (algebraic) automorphic representation ...
9
votes
2answers
2k views

p-adic L-functions

For modular forms, it is known that you can construct p-adic L-functions by interpolating (p-power conductor) twists of their associated L-functions at special values. Similarly, Kubota-Leopoldt's ...
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 ...
9
votes
2answers
856 views

Automorphic forms and Galois representations over imaginary quadratic fields: generalizing Taylor's theorem?

Let $K/\mathbf{Q}$ be an imaginary quadratic field, with $\sigma \in \mathrm{Gal}(K/\mathbf{Q})$ a generator. Suppose $\pi$ is a cuspidal automorphic representation of $GL_2 / K$ with central ...
9
votes
1answer
406 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: ...
9
votes
0answers
315 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 ...
8
votes
1answer
531 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?
8
votes
2answers
341 views

central/critical/special values of L-functions terminology

I have a question about the terminology for special values of L-functions. Is the following a correct description of standard usage: Suppose L(s) is an L-function which satisfies a functional ...
8
votes
1answer
809 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) = ...
8
votes
1answer
370 views

Modularity of higher dimensional abelian varieties

In another question I asked about strategies for giving an effective version of the Shafarevich conjecture for abelian varieties over $\mathbb{Q}$. For elliptic curves, one can give a proof using ...
8
votes
1answer
605 views

Finite dimensional automorphic representations of a definite quaternion with prime discriminant and Hecke action

Before stating the questions that I have, which are very specific and probably not so interesting to someone who has never thought about these things, I need to introduce some notation. Let $p$ be ...
8
votes
1answer
184 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 ...
8
votes
1answer
291 views

Atkin–Lehner operator for GL(3)?

Let $f$ be an automorphic form for $\Gamma_0(N)\subset SL(3,\mathbb{Z})$. $\Gamma_0(N)=(a,b,c;d,e,f;g,h,i)\in SL(3,\mathbb{Z})|g=h=0(mod N)$ Is there any Atkin-Lehner operator for $\Gamma_0(N)$ ...
8
votes
0answers
153 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
201 views

Regularity assumption in the simple trace formula

In the simple trace formula of Deligne Kazhdan one assumes that the test function is supported at the elliptic regular elements at one place and is a supercusp form at another place. Why can't one ...
8
votes
0answers
194 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 ...