The automorphic-forms tag has no wiki summary.

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### 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**

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**2**answers

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### 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**

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**3**answers

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### 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**

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**6**answers

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### 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**

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### 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 ...

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### 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**

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### 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**

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### 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**

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**1**answer

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### 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**

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### 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**

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**1**answer

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### 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**

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### 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**

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**2**answers

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**

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**3**answers

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### 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**

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### 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**

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**3**answers

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**

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**4**answers

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### 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**

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**1**answer

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**

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**3**answers

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### 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**

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**3**answers

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**

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### 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**

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**4**answers

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### 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 ...

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### 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 adĂ¨les of $F$, $\mathbf{A}_f$ the finite adĂ¨les of $F$. Fix a maximal compact ...

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### 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 ...

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**2**answers

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### 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**

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**2**answers

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**

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**1**answer

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### 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**

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**2**answers

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**

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### 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**

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**1**answer

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### 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**

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**1**answer

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**

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**1**answer

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### 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 $d/n$ ...

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### 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**

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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**

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**2**answers

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### 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**

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**1**answer

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 ...

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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 ...

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### 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:
...

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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**

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**1**answer

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**

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**2**answers

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**

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**1**answer

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**

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**1**answer

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### 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**

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**1**answer

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 ...

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**1**answer

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**

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**1**answer

291 views

### Atkinâ€“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 $\Gamma_0(N)$ ...

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### 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**

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**1**answer

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 ...

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### 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 ...

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### 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 ...