Questions tagged [automorphic-forms]

Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups.

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How to compute that the volume of $Z(\mathbb{A})\backslash \frak{S}$ for a Siegel set is finite?

I found it quite hard to find a complete reference for the (adelic) reduction theory of algebraic groups used in automorphic representations. I want to show that for $G=GL(n)$ over a number field $F$, ...
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Reference request: multiplicative group of a central simple algebra, their reductivity and parabolic subgroups

During my study of the theory of automorphic forms and $L$-functions, I never found any literature dealing with the following: Suppose $D$ is a central division algebra over a local or global field $F$...
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Compute the Fourier expansion of adelic Eisenstein series associated to the classical holomorphic Eisenstein series.

For each place $v$ of $\mathbf{Q}$, define $\Phi_v:(\mathbf{Q}_v)^2\to\mathbf{C}$ by $$ \Phi_v(x,y)=\begin{cases} \mathbb{I}_{\mathbf{Z}_v}(x)\mathbb{I}_{\mathbf{Z}_v}(y)&\text{if $v<\infty$},\\...
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A measure theoretic problem related to induced representations

I met such a rather concrete measure theoretic problem while dealing with induced representations of $p$-adic groups. So let $G$ be a unimodular locally compact group, with $P$ its closed subgroup (...
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A convergence lemma for adelic zeta function in automorphic forms

I'm reading Godement-Jacquet's classic Zeta functions of simple algebras (1972, Springer). On page 153, the first line: " We also $\textbf{take for granted}$ the following lemma (numbered 11.3)......
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Intuitive understanding of Tamagawa measure and its relationship between local measures?

Weil's book Basic Number Theory, (1973, second edition) pp. 113 mentioned the Tamagawa measure on $k_\mathbb{A}$, where $k$ is a global field and $k_\mathbb{A}$ its ring of adeles (an old fashioned ...
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Writing the automorphic form $\phi_f$ of a newform $f$ as a pure tensor

Let $\mathbb{A}$ be the adele ring of $\mathbb{Q}$. Let $f\in S_k(\Gamma_0(N),\chi)$ (possibly a newform) and $\phi_f:\text{GL}_2(\mathbb{Q})\backslash\text{GL}_2(\mathbb{A})\to \mathbb{C}$ be its ...
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Smooth functionals in the contragredient representation of a locally profinite group

I'm reading Bump's book Automorphic Forms and Representations and getting stuck in Section 4.2 about contragredient representation of smooth representations. Setup: Let $G$ be a locally profinite ...
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Whittaker at Archimedean Test vector

Let $\pi$ be a generic irreducible Casselman Wallach representation of $GL_n(\mathbb{R})$. Let $\phi$ be a test vector for the archimedean local L function $L(s,\pi)$. Is there an explicit expression ...
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Integrating an additive character over a local field

Let $F$ be a non-Archimedean local field, $\psi$ a non-trivial additive character of $F$. Let $\mathfrak{o}$ be the ring of integers of $F$, and $\mathfrak{p}$ be the maximal ideal of $F$. Endow $F$ ...
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Application of Selberg Pre-Trace formula

I am trying to solve the following problem: Prove the following estimate: $$\sum_{\vert t_i\vert<T}\vert u_j(z)\vert^2\ll T^2$$ by choosing appropriate $h$ in the Selberg's (pre)-trace formula. I ...
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Why are Rankin-Selberg convolutions different when $n=m$?

Let $\pi$ an automorphic cuspidal representation on $GL_n(\mathbb{A})$ (and similarly $\pi'$ on $GL_m$). For $m=n-1$, it is standard to introduce the Rankin-Selberg L-function as (the gcd of the ...
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The notions of $K$-finite vectors in admissible representations

Let $G$ be a real Lie group and $K$ be a maximal compact subgroup. Let $(\pi, V)$ be a Hilbert space continuous representation of $G$. It can be restricted to $K$ as a representation of $K$. For each ...
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About holomorphic functions on the upper half plane with respect to SO(2).

Let $\mathbb{H}$ be the upper half plane. For $g=\left(\begin{array}{cc}a&b\\c&d\end{array}\right) \in SL_2(\mathbb{R})$ and $z \in \mathbb{H}$, set $J(g,z)=cz+d$. Let $SO(2)$ be the special ...
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*Why* the converse theorem (conjecture) should be true?

Currently, the following $\mathrm{GL}(n)$ converse theorem due to Cogdell and Piatetski-Shapiro is known: For an admissible irreducible representation $\pi$ of $\mathrm{GL}(n, \mathbb{A})$, $\pi$ is ...
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Relations between different zeta functions for a simple algebra

I'm trying to understand the classical works of Eichler, Shimura and many others (especially Shimizu and Tamagawa's annals papers) on the "classical" (I'm a newcomer and I'm not sure whether ...
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Explicit definition of the cusps of a congruence subgroup of the symplectic group

I'll begin by defining the notion of a cusp of a congruence subgroup of $\textrm{SL}_2(\mathbb{Z})$: $\textrm{SL}_2(\mathbb{Z})$ has a natural action on the compactified upper half complex plane $\...
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Background behind Eichler's discovery of the relationship of a modular form with an elliptic curve

In Fraenkel's Love and Math (and Richard Taylor's Modular Arithmetic IAS Post https://www.ias.edu/ideas/2012/taylor-modular-arithmetic), specifically in Chapter 8 Magic Numbers, page 88., Fraenkel ...
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Why does the "automorphic factor" exist?

In the theory of automorphic forms, functions we consider are not totally invariant (why? is it because otherwise there would be too few of them, like none except constants?), but "twisted" ...
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Reference request: Katz modular forms modulo $p$ at cusps vs classical modular forms modulo $p$ at cusps

I, like the author of this post, am severely lacking the background to make the connection between reducing modular forms' $q$-expansions modulo $p$ at various cusps, and $q$-expansions of modular ...
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Compactness of automorphic quotient - quaternion group and $\mathrm{GL}_{2}$

Let $\mathbb{A} = \mathbb{A}_{\mathbb{Q}}$ be an adele ring over $\mathbb{Q}$ and $G_{1} = \mathrm{GL}_{2}$, $G_{2} = \mathrm{Res}_{D/\mathbb{Q}}\mathrm{GL}_{1}$ be two groups over $\mathbb{Q}$, where ...
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Classification of Representation of $GL(2,\mathbb{R}^+)$

$GL^+(2,\mathbb{R})$ is the notation for positive non-zero determinant square matrices of order two. I am reading Bump's book on Automorphic Forms and Representations. One of the major part of second ...
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F-split $SO(2)$ is isomorphic to $GL(1)$?

Let $F$ be a number field and $\chi$ is a character of $\mathbb{A}^{\times}/F^{\times}$. In some paper, it is written that we can consider $\chi$ as a representation of $F$-split $SO(2)$. I am ...
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Explicit Jacquet-Langlands correspondence

Jacquet-Langlands correspondence gives a 1-to-1 correspondence between automorphic forms on $\mathrm{GL}_{2}(\mathbb{Q})$ and automorphic forms on $\mathrm{GL}_{1}(D)$, where $D$ is a division algebra ...
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Bijection between two orbit's spaces

Take $\mathbb A_\mathbb Q$ the adele ring of $\mathbb Q$ and let $$\vert \cdot\vert:\mathbb A_\mathbb Q\to \mathbb R_{>0}, x=(x_p)\mapsto \prod_{p\leq \infty}\vert x_p\vert_p$$ the absolute value ...
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About definition of the global Hecke algebra and the automorphic form

Introduction I am trying to understand various definitions (which I hope them equivalent) of automorphic forms. My main problem is to describe with the global Hecke algebra rather than the group and ...
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Properties of the Galois Representation Attached to a Weight 2 Cusp Form

The paper here posits on page 86 that it is Shimura who proved in "Introduction to the Arithmetic Theory of Automorphic Functions" that for a prime $p$, the $p$-adic Galois representation ...
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Determining what "moduli scheme" means in Katz's paper in regards to his Corollary

I'm trying to "decode" Corollary 2.6 of Katz's text in the case $m = 1$. A shortened background: Fix a prime $p > 5$ and $N \in \mathbb{N}$ prime to $p$. Let $F := \mathbb{F}_{p}(\zeta_{N}...
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A finite-volume problem in the adelic group of a global field

I am reading Zeta functions of simple algebras by Roger Godement and Hervé Jacquet. In p139-140 they introduce a version of the theory of reduction. To get the finiteness of the Siegel domain module ...
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Problem regarding $PSL_2 (\mathbb{Z})$ invariant function

Let $\Gamma = PSL_2(\mathbb{Z}) $ and f be a $\Gamma$ invariant function on the upper half plane. Further let $F$ be the standard fundamental domain of $\Gamma$ and $F^{’ }=gF$ be the image of $F$ ...
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An identity regarding the product of two $K$ Bessel functions.

In Goldfeld’s book on automorphic forms on $GL(n)$, particularly while discussing the Rankin Selberg method for $GL(2)$ he write a formula for the “inner product” of two $K$ Bessel functions (shown ...
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Factor the global induced representation into the restricted tensor product of local induced representations in Bump's book

Some notations: $F$ a global field with places $v$. $\chi_1$, $\chi_2$ two Hecke characters $A_F^\times/F^\times\to\mathbf{C}^\times$. $K$ the compact maximal subgroup $\prod_{v\text{ real}}O(2)\...
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What are the prerequisites for studying Automorphic Representation Theory?

I am interesting in getting into Automorphic Representation Theory but am unsure where to start. It seems like there are a lot of prerequisites. What are the main areas (above abstract algebra) that ...
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Relations between $M_k(\Gamma_0(N))$ and $M_k(\Gamma_0(L,M))$, and Eisenstein Series.

Consider the following congruence subgroups $$\Gamma_0(N):=\Big\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix}\in\text{SL}_2(\mathbb{Z})\; \Big| \; c\equiv0\text{ (mod }N)\Big\}$$ and $$\...
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Is the Hilbert space $L^2$(G\ $\mathbb{H}$) separable for G a subgroup of $SL_2(\mathbb{R})$ acting on upper half plane?

I was asked by a friend this question: Suppose G is a subgroup of $SL_2(\mathbb{R})$ acting on the upper half-plane, such that the boundary of the fundamental domain of the action has zero measure. Is ...
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Point Pair invariants

I have been reading Iwaniec's spectral theory of Automorphic forms, and in one of its definition, its defined that a function $k: \mathbb{H}× \mathbb{H} \rightarrow C$ is point pair invariant if $k(gz,...
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Reference for Langlands functoriality conjecture view towards classical examples

I want to know if there's any good reference on Langlands functoriality conjecture which provides connection with classical examples. What I have in my mind are followings: Classical Rankin-Selberg (...
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Notion of a Jacobi group covariant differential equation

In D. Zagier's paper Modular forms and differential operators it is stated that, ... every modular form satisfies a non-linear third order differential equation with constant coefficients; in another ...
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Hilbert Modular Forms defined on $SL_2$ or on $GL_2^+$?

I am starting to read up on Hilbert Modular Forms and I am a bit confused with the definition. I am looking at multiple references and I am seeing some definitions that define Hilbert Modular Forms to ...
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Vanishing order of a modular form at an irregular cusp: why is it treated like a valuation?

I am studying Diamond and Shurman’s book on modular forms and I do not understand the computation of dimension formulas for odd weights (page 90). To make the setting more precise, let $\Gamma$ be a ...
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Why is a theory of newforms in half-weight difficult? Why are $U$-operators used instead of $V$-operators?

Integer weight case: Modular forms in $S_{2k}(N) := S_{2k}(\Gamma_{0}(N))$ can come from lower levels, and we'd like to know when that happens. This is where we call a modular form $f$ "old" ...
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Action of special orthogonal group on Upper half complex plane.

When group $G=SL_{2}(\mathbb{R})$ acts on upper half plane I am able to find the stabiliser subgroup of $i$ is nothing but special orthogonal group $\{ \begin{bmatrix} \cos x & \sin x \\ -\sin x &...
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Idea of computing local zeta integrals

I'm reading Gelbart's classic (Automorphic forms on adeles groups) in a seminar. And now we're dealing with section 6.C, Jacquet-Langlands's work using local-global zeta integral as follows: $$ Z(W, s,...
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On Fourier coefficients of Bianchi modular forms, l-ordinary

Let $f\in S_2(\Gamma_1(N))$ be a Hecke eigenform and $\ell$ a prime number does not divide $N$. Let $a_f(\ell)$ be the $\ell$-th Fourier coefficient of $f$. Then $a_f(\ell)$ is is called $\ell$-...
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The automorphic green function

Denote by $\Delta$ the Laplacian on $SL_2(\mathbb{R})$ with the standard Haar measure. Denote by $A_{\lambda}$ the operator $$\dfrac{1}{\Delta^2 - \frac{1}{4}\frac{d^2}{d\theta^2} + \lambda}.$$ The ...
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What is an odd Maass form

A Maass form for a congruence subgroup $\Gamma$ of $SL(2, \mathbb Z)$ is a complex valued smooth function $f$ on the upper half plane $\mathbb{H}$ such that $f$ is $\Gamma$-invariant $f$ is ...
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Is it possible for $f(z)$ and $f(\sigma z)$ to both be Hecke eigenfunctions?

Let $f(z)$ be an automorphic/modular form on $\Gamma_0(p)/\mathbb{H}$, where $p$ is some prime. I know that the Hecke operators $T_n$ act on this space of functions whenever $(n,p)=1$. Assume that $f(...
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A Hecke-Maass eigenbasis for the space of Maass cusp newforms

I heard that the space of Maass cusp newforms on $\Gamma_0(N)/\mathbb{H}$ has a basis of Hecke eigenforms. Would anyone happen to know of a reference of this fact? Or, even better, how to prove it?
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Petersson formula with powers

The classical Petersson formula gives a formula for $$\sum_{c>0} \frac{S(m, n, c)}{c} J(m, n, c)$$ for $J$ an explicit Bessel function. Is there a similar formula existing for different weights or ...
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Where do automorphic Maass forms come from?

I know the Langlands program for GL2/SL2 gives some hints as to the origins of automorphic representations, and that some cases of the correspondence have been completely classified or reasonably ...
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