An automorphic form is a well-behaved function from a topological group G to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of the topological group. Automorphic forms are a generalization of the idea of periodic functions in ...

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stationary phase method in analytic number theory

I hope someone can tell me something about the error term in the formula calculating the oscillatory integral like $\int_a^b g(x)e(f(x))d x$. Specially, the exact formula on page 114 of M. Huxley's ...
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264 views

What is a “cohomological type” automorphic representation?

Sorry for asking such a question. This is supposedly a well known definition for experts. However as an non-expert, I tried (very hard) google, Mathscinet etc, but still couldn't find a satisfactory ...
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243 views

Is $(G,K)$ a strong Gelfand pair?

Let $F$ be a $p$-adic field with ring of integers $\mathcal{O}$. When $G={\rm GL}_n$, it is a classical result that $(G(F),G(\mathcal{O}))$ is a Gelfand pair. Is it actually a strong Gelfand pair? I ...
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1k 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?
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349 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 ...
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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 ...
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321 views

Classical Lower Bound of L(1) assuming GRH

Let $L(s)$ be an automorophic $L$-function with conductor $C$ defined by Iwaniec and Sarnak. What's the classical lower bound of $L(1)$ assuming Riemann Hypothesis? And what's the reference? Is ...
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370 views

A frustrating cohomology class on the moduli of abelian surfaces

Here's a very frustrating question that I have been stuck on for some time. I believe that my question could fit in a general framework of what happens when you restrict $L^2$-cohomology classes on a ...
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1answer
147 views

Are supercuspidal reps of GL(2) uniquely determined by the rootnumber

Let $\pi, \pi'$ be a unitary, irreducible, supercuspidal representations of $GL_2(F)$. Does an equality of roots numbers $\epsilon(\pi, \psi, s) = \epsilon(\pi', \psi, s)$ for all $s \in \mathbb{C}$ ...
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124 views

Bessel function for $GL_3(\mathfrak{R})$?

In the $GL_2(\mathfrak{R})$ case, assume that $\pi$ is an irreducible unitary representation and $W_{\pi}(g)$ is the Whittaker functional associates with $\pi$. Then there is a Bessel function $j_{\pi}...
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569 views

subconvexity problem for $GL(3) × GL(2)$ $L$-function without involving in symmetric lift

A question in study of subconvexity topic puzzles me for a long time, which mabe a stupid question for many experts. I really wish someone to help me out, and any advice will be highly appreciated. ...
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6answers
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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 $...
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83 views

Sign of the functional equation of L function and Shimura lift

I would like to know what happens to the root number of a half integral weight automorphic form (holomorphic or not), i.e. the sign of the functional equation of its L-function when we apply the ...
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862 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 ...
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133 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 ...
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328 views

How to translate the representation theory of semisimple to reductive groups?

I am aware of the following question: Definitions of Reductive and Semisimple Groups So let me phrase a precise question: Is there a standard technique by which one can translate the unitary/...
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375 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\; \phi)=\zeta(...
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251 views

Restriction map between spaces of automorphic forms

Hello, Let $H \subset G$ be reductive groups defined over $\mathbb{Q}$. I consider the spaces of automorphic forms of $G$ and $H$. One has a restriction map from the space of automorphic forms of $G$ ...
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104 views

Theta lift to 1-dimensional vector space.

Hi! My question is very simple and it is about the theta lift of unitary group in global situation. Let $E/F$ be a quadratic number fields and $V,W$ be an $n$-dimensional and $1$-dimensional ...
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547 views

Cuspidal automorphic representations as the space of $K$-finite vectors in a unitary cuspidal automorphic representation.

The definition I know of for a cuspidal automorphic representation of, say, $G=\mathrm{GL}_2$ over a number field $F$ (relative to a choice of compact open subgroup $K_f$ of $G(\mathbf{A}_F^\infty)$ ...
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248 views

On the Cartan decomposition of unitary group

Hello. I have some question on Cartan decomposition of unitary group, especially $U(2)$. I am interested in local situation, that is p-adic or archimedian. Let $F$ be a local field and $E$ be its ...
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178 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 =SL_2(\mathbb{R}...
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384 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 $GL_2(...
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No Exceptional Eigenvalues of Weight 1/2 Maass Forms on $\Gamma_0(4)$?

Some colleagues and I were wondering if there is a citation out there which shows there are no exceptional eigenvalues, $\lambda$, of classical weight 1/2 Maass forms on $\Gamma_0(4)$, which is to say ...
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478 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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111 views

Integral conjugacy vs. Rational conjugacy

Let $G$ be an algebraic group over a field $F$. Let $g\in G(F)$, and write $C(g)$ for the centralizer of $g$ in $G$. Conjugacy over $F$ is of course not necessarily the same thing as conjugacy over an ...
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555 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 ...
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150 views

Off critical line zeros for half integer weight $L$-functions

Let $f(z) = \sum_{n=1}^\infty A(n)n^{\frac{k-1}{2}}e(nz)$ be a modular form of weight $k$ for a half integer $k$. Put $$L(s,f) = \sum_{n=1}^\infty \frac{A(n)}{n^s} $$ to be the $L$-function. Further ...
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142 views

On the absolute convergence of the local-zeta integral.

Though I am in a situation considering only local-zeta integral, to explain my question briefly, let me ask it in quite general form. Let $f(s,g)$ be a two variable smooth good (in a suitable sense) ...
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82 views

Non vanishing and cuspidality of the theta lift of trivial representation.

Hi! Let E/F be a quadratic number field extension. Then we make some hermitian and skew hermition vector spaces and define unitary group on it.(namely U(1) and U(3)) Then, I am wondering whether the ...
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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 ...
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102 views

On the explicit formula of the height function occuring on the doubled Weil representation.

Hi! I am wondering the exact formula of height function of $GL(n)$ which occurs in the doubling Weil representation. To be more precise, let me introduce the basic setting for this. Let $F$ be the ...
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383 views

Is Eisenstein series not identically zero

How does one prove that an Eisenstein series (adelically formulated as in the book of Moeglin-Waldspurger) is not identically zero? Namely how does one prove that the sum $\sum_{\gamma\in P(k)\...
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2answers
397 views

A question on twisted L-function

Hi! This is my first question here. In studying automorphic form, I am wondering the relation of critical L-values of some representation and its twisted representation by a character. For example, ...
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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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345 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 $PGL_2(\mathbb{A})$. ...
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257 views

Hecke eigenvalue at p and at p^k

I am interested in the relationship between the Hecke eigenvalue at $p$ and at $p^k$ for $k \geq 2$ in the unramified and ramified situation for modular/Maass forms. More precisely, I know from a ...
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102 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?
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239 views

Arthur-Clozel Prop 3.1 for Function Fields?

The subject says it all. I would like to know if Proposition 3.1 in Arthur-Clozel's book on the trace formula holds for local fields of positive characteristic. Thanks! EDIT: Here is Prop 3.1 of ...
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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.
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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: http://publications....
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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 \pi_{...
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Refining the moderate growth condition on automorphic forms

Let $G$ be a simple algebraic group defined over $\mathbb Q$. In their Corvallis article (automorphic forms and automorphic representations), Borel and Jacquet define an automorphic form to be a ...
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A stupid question on automorphic l-function

This may be a silly question for experts in this area. But I am really suffering for not being able to compute local-L function of some automorphic representation. So, I post it hoping some ...
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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 ...
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258 views

On the theta lift and its L-function

I am wondering how the relation is between of the automorphic L-function and its lift's. More precisely, Let $E/F$ be a quadratic extension of number fields. Let $W$ be a hermitian space over E of ...
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236 views

On the restriction of cusp. irr. representation and period.

Let's consider only global case. Let $G_n$ be classical algebraic group over global # field (eg, $GL(n),SO(n), U(n)$...) and let $\pi_n$ be its irr. cusp. reps of $G_n$. Then we can define the ...
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270 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 ...
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1answer
174 views

Global square integrability ensures local sq. integrability?

This might be a stupid question for expert in this area. I am considering automorphic representation of algebraic group. In studying it, local tempered, local square integrable representations ...
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413 views

a question about the proof of analytic continuation of Eisenstein series for GL(2)

I'm reading Gelbart and Jacquet's paper 'forms of GL(2) from the analytic point of view', and was confused at a point in the proof of analytic continuation of Eisenstein series. On the top of page 232,...