Questions tagged [automorphic-forms]

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 Euclidean space to general topological groups.

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The inclusion of Siegel sets of the general linear groups

In some paper, it is written that none of Siegel set of $GL_{n}$ is not contained in any Siegel set of $GL_{n+1}$. I don't understand this because I think it should hold. To explain my question more ...
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Question on Iwasawa decomposition of unitary groups over adele ring

Let $E/F$ be a quadratic extension of number fields and $V,\langle,\rangle$ is a hermition vector space over $E$. Let $\mathbb{A}$ be the adele ring of $F$. Assume that there is a hermitian line $e \...
Andrew's user avatar
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The product of $Z(\mathfrak{g})$-finite functions is also $Z(\mathfrak{g})$-finite?

Let $G$ be a classical group defined over $\mathbb{Q}$. Let $\mathfrak{g}$ be the Lie algebra of $G(\mathbb{R})$ and $U(\mathfrak{g}_{\mathbb{C}})$ its universal enveloping algebra of $\mathfrak{g}_{\...
Andrew's user avatar
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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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Powers of automorphic Eisenstein series

Let $G$ be a reductive group defined over $\mathbb{Q}$. Let $P$ be a standard parabolic subgroup of $G$ with Levi decomposition $$P = MN.$$ We denote by $R_{disc,M}$ the discrete spectrum of $M$. Let $...
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Functoriality for compactifications of locally symmetric spaces

Let $X$ be a symmetric space associated to an algebraic group $G$ defined over $\mathbb{Q}$ and $G(\mathbb{R})$ acts on $X$ from the left. Let $\Gamma \subset G(\mathbb{Q})$ be an arithmetic subgroup ...
random123's user avatar
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Formal degree of discrete series representations

Let $G$ be a locally compact unimodular group. A continuous irreducible unitary representation $\pi$ of $G$ is said to be a discrete series if its matrix coefficients (defined by $\xi^\pi_{v,w} : g \...
Desiderius Severus's user avatar
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Relation between $\xi$-cohomological and discrete series

Sometimes, results on automorphic representations are available only under local assumptions. Typically, one could require the representation to be a $\xi$-cohomological cuspidal representation, and I ...
Desiderius Severus's user avatar
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Truncation and weighted orbital integrals in hyperbolic term of trace formula for $\mathrm{GL}(2)$

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PGL{PGL}$I am looking at Gelbart--Jacquet's article in the first Corvallis volume (the article entitled Forms of $\GL(2)$ from an analytic point of ...
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Contemporary introduction to Godement-Jacquet "Zeta functions of simple algebras"

The question is in the title: The book Godement-Jacquet "Zeta functions of simple algebras" is from 1971. Has there ever been a textbook introduction to this material, or at least part of it?...
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Holomorphic automorphic/cusp forms on real Lie groups

An automorphic form on a real Lie group $G$ for a discrete subgroup $\Gamma$ is a function $f:G\to\mathbb{C}$ with some properties (see Borel’s definition in Proceedings of Symposia in PURE ...
Jun Yang's user avatar
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Statement of classical Ramanujan-Petersson conjecture

I'm preparing for an expository talk on some topics in the representation theory of reductive p-adic groups, including tempered representations and Whittaker models, and as motivation I wanted to ...
David Schwein's user avatar
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From Maass forms to automorphic forms of $SL_2(\mathbb{A})$

I'm learning basic stuff about automorphic forms, please, if anything I say is not true, excuse me. Let $X = \Gamma\setminus \mathcal{H}$ be a modular curve and let $f(\tau)$ be a Maass cusp form. ...
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Finiteness of the volume of $G(F) \backslash G(\mathbb A)$

Let $G$ be a semisimple algebraic group over a number field $F$ with trivial center. Let $\mathfrak S \subset G(\mathbb A)$ be a Siegel domain (defined in terms of a given maximal split torus and ...
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Eigenvarieties and functoriality

In Langlands' review of Hida's book "$p$-adic automorphic forms on Shimura varieties", he discusses a nexus of 4 areas of modern number theory: automorphic representations, motives, spaces ...
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What are the known number-theoretic functions, that are related to "the number of ideals of norm $n$, that belong to the class $[c]$"?

Let $L$ be a number field, $\mathcal{O}_L$ its ring of integers, and $\mathcal{Cl(O}_L)$ its ideal class group. Let's fix an arbitrary class $[c] \in \mathcal{Cl(O}_L)$. By $r(n)=r([c], n)$, I mean ...
Davood Khajehpour's user avatar
11 votes
1 answer
588 views

Modularity of higher genus curves

The modularity conjecture for elliptic curves over number fields is well known, and indeed, is a theorem for all elliptic curves over $\mathbb{Q}$, and at least potentially, over any CM field. What ...
Nimas's user avatar
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Jacquet-Langlands correspondence for Norm-1-units

Let $M$ be a quaternion algebra over $\mathbb{Q}$, ramified at the set of primes $S$. Let $\pi=\bigotimes_p \pi_p$ be an irreducible, infinite-dimensional subrepresentation of $L^2(M_{\mathbb{A}}^\...
Julien's user avatar
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What's the motivation for the $3$ appearing in Iwaniec and Kowalski's definition of the analytic conductor?

In their book Analytic Number Theory, Iwaniec and Kowalski, on page 95, define the analytic conductor by the following formula: $\displaystyle{{\frak{q}}_{\infty}(s)=\prod_{j=1}^{d}\left(\vert s+\...
Sylvain JULIEN's user avatar
6 votes
1 answer
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Two definitions of automorphic forms on Lie groups

My question is the about the equivalence of two definitions of automorphic forms on a semisimple Lie group. The most common definition of automorphic forms on a semisimple Lie group $G$ with respect ...
Jun Yang's user avatar
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Spectral decomposition of product of modular functions

The eigenfunctions of the Laplacian on $SL(2,\mathbb Z)\backslash \mathbb H$ are known to be given by three types: the constant function, the real analytic Eisenstein series (which come in a ...
nathan benjamin's user avatar
14 votes
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Bound for $GL(3)$ symmetric square

Let $\pi$ be an automorphic representation of $GL(3)$ over a number field. Let $a_n$ be the coefficients of $L(s, \pi, \mathrm{sym}^2)$. Do we know if $$\sum_{n>0} \frac{|a_n|}{n^s}$$ and $$\sum_{n&...
Desiderius Severus's user avatar
17 votes
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998 views

Automorphic forms and coherent cohomology

Why is it (and what does it mean) that automorphic forms do not contribute in the coherent cohomology of Siegel modular varieties parametrizing abelian varieties of dimension $d>2$ (see section 7 ...
Anton Hilado's user avatar
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Conway big picture for congruence subgroups of $\mathrm{SL}_3(\mathbb{Z})$

I saw in Conway’s paper "Understanding groups like $\Gamma_0(N)$" that the so-called Big Picture can give simple interpretations for important objects in number theory, such as Hecke ...
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Why admissible representations?

In the theory of automorphic forms we often right away reduce the study to admissible representations, and I wonder how much everything breaks when not. If I understood well, admissible ...
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What is the meaning of the $L$-group?

Langlands' functoriality conjecture predicts that to a suitable homomorphism of $L$-groups $$ \psi : {}^LG \to {}^LH $$ there should be a transfer of automorphic representations from $G$ to $H$. For ...
Tian An's user avatar
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What is a map for the representation theory of reductive groups?

I have finished learning about linear algebraic groups (minus their representation theory) and the associated algebraic structures (root data, root systems, etc.), and will next attempt to summarize ...
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Computing explicit matrix coefficients

I would like to understand in a more explicit way the Fell topology on unitary duals, that is to say the convergence of matrix coefficients of local representations. If I consider a local ...
Desiderius Severus's user avatar
2 votes
1 answer
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Quotient space by discrete group and $L^2(\Gamma\backslash \mathcal{H})$

I'm reading Daniel Bump's "Automorphic forms and representations" chapter 2, and in the book they define an integral over $\Gamma\backslash\mathcal{H}$ (here, $\Gamma$ is a discontinuous ...
hong's user avatar
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Maass--Selberg for any Eisenstein series on higher rank

Does there exist a Maass--Selberg relation for any Langlands Eisensein series on $\mathrm{GL}(n)$? By any I mean an Eisenstein series which is induced from any standard parabolic with any discrete ...
Subhajit Jana's user avatar
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Automorphic representation of weight 3 eigenforms

Let $f$ be a weight 3 eigenform with rational Fourier coefficients. As shown by Elkies and Schutt, $f$ is associated to a singular K3 surface over $\mathbb{Q}$. A construction of Shioda and Inose ...
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Are there infinitely many L-rigs?

$\DeclareMathOperator{\Q}{\mathbb{Q}}$Call "L-rig" any class $\mathcal{L}$ of L-functions of automorphic representations of $\operatorname{GL}_{n}(\mathbb{A}_{\Q})$ for some $n$ belonging to ...
Sylvain JULIEN's user avatar
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0 answers
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Relation between automorphic representations and Laplacian eigenvalues

Let $G=SL(2, \mathbb{R})$. I would like to understand how and why automorphic forms correspond to automorphic representations, and how general is this fact (for other groups). Let $\Gamma$ be a ...
Wirdspan's user avatar
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Principal series representation of $SL(2,\mathbb{R})$ restricted to principal congruence subgroup

Given a principal congruence subgroup $\Gamma(N)$ of $SL(2,\mathbb{R})$, since $\Gamma(N)$ is free, consider a probability distribution $\mu_1$ of a simple random walk on $\Gamma(N)$ and consider its ...
user158970's user avatar
4 votes
0 answers
384 views

What is a Shimura variety?

It so look like that Shimura variety is a modular space that contains classes of zeros of modular functions that have certain symmetry that have structure of Hodge type. Is this true?
Wahadti Vitalis's user avatar
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Polyharmonic Maass forms are automorphic forms on $\mathrm{SL}_2(\mathbb{R})$

Let $G=\mathrm{SL}_2(\mathbb{R})$, $K=\mathrm{SO}(2)$, and $f$ be a holomorphic modular form of weight $k$ for $\Gamma$ a Fuchsian group of the first kind. In Borel's book, 'automorphic forms on $\...
LWW's user avatar
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1 answer
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How did Gauss characterize the metrical relations in the uniform (4 4 4) tiling of the hyperbolic unit disk?

My purpose is to verify an historical hypothesis I have on Gauss's tesselation of the unit disk as described in John Stilwell "Mathematics and its history". Looking at the relevant pages in ...
user2554's user avatar
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8 votes
1 answer
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Characterization of automorphic discrete spectra

I recently learned about automorphic spectral decomposition from the book "Spectral decomposition and Eisenstein series" by Moeglin and Waldspurger. (Let me call it M-W) I have a question ...
Aut's user avatar
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9 votes
1 answer
670 views

The cohomology of modular curves as a module over the Galois group

Consider the modular curve $\pi: X(N) \to X(1)$ where this map has Galois group $G = PSL_2(\mathbb Z/N\mathbb Z)$. In particular, $G$ acts on the singular cohomology $H^1(X(N),\mathbb Z)\otimes \...
Asvin's user avatar
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11 votes
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Representation theory of $\operatorname{GL}_2(\mathbb Z/n\mathbb Z)$

Is there a nice reference for the finite dimensional (characteristic 0) representation theory of $\operatorname{GL}_2(\mathbb Z/n\mathbb Z)$ and $\operatorname{PGL}_2(\mathbb Z/n\mathbb Z)$ for ...
Asvin's user avatar
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7 votes
0 answers
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Theta Function Associated to Kummer Lattice

This is something which I feel must be out in the literature somewhere, but I have been unable to find anything. If we let $\text{Km}(A)$ be the Kummer $K3$ surface associated to an abelian surface $A$...
Benighted's user avatar
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1 vote
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Does the standard arithmetic subgroup of a closed $\mathbb{Q}$-algebraic groups have non-trivial $\mathbb{Q}$-characters?

I am trying to understand the Borel-Harish Chandra theorem about arithmetic subgroups being lattices. Suppose $G$ is an algebraic group inside $GL_n(\mathbb{C})$ such that it is definable as a zero ...
Breakfastisready's user avatar
8 votes
2 answers
628 views

A question related to Hilbert modular form

This is a question related to Hilbert modular forms. Let $\mathbb{K}=\mathbb{Q}(\sqrt D)$ be an imaginary quadratic field with discriminant $D<0$ and $\zeta (\text{mod } m)$ a Hecke character such ...
user15243's user avatar
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4 votes
1 answer
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anti-holomorphic Hilbert modular forms as global sections

The classical definition of Hilbert modular cuspforms as given in say, Hida's "On $p$-adic Hecke algebras for $\mathrm{GL}_2$ over totally real fields", defines them as holomorphic functions ...
Leray Jenkins's user avatar
2 votes
1 answer
260 views

Question on the residual representation

Let $G=SO_n$ and fix a borel subgroup $P_0$ of $G$. Let $P=MN$ be a standard maximal parabolic subgroup $G$ and $\sigma$ a cuspidal representation of $M$ Consider the normalized parabolic induced ...
Monty's user avatar
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1 vote
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Question on induction of unramified representations

$\def\anonabs{\lvert\cdot\rvert}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Ind{Ind}\DeclareMathOperator\SO{SO} $Let $F$ be a $p$-adic local field of characteristic zero. Let $\chi$ be an ...
Monty's user avatar
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3 votes
0 answers
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Question on the proper sub-representation of induced representation

$\DeclareMathOperator\Ind{Ind}$Let $G$ be a reductive group over a $p$-adic local field $F$, and $P=MN$ a parabolic subgroup. Let $\sigma$ be an irreducible representation of $M(F)$ and consider its ...
Monty's user avatar
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23 votes
2 answers
832 views

Moments of Plücker coordinates on complex Grassmannian

Consider the Grassmannian $Gr(k,N)\simeq U(N)/(U(k)\times U(N-k))$ which parametrizes $k$-dimensional subspaces of $\mathbb{C}^N$. Let us put on it the $U(N)$-invariant probability measure. Let $\...
Abdelmalek Abdesselam's user avatar
7 votes
0 answers
283 views

The Fontaine Mazur conjecture for $\text{GL}_1$ over $\mathbf{Q}$

The Fontaine-Mazur cojectures says that if $\chi : \text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \mathbf{Q}_p^\times$ is a character which is unramified almost everywhere and de Rham at $p$ then it ...
Jdoe's user avatar
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3 votes
1 answer
252 views

Level vs. conductor of a supercuspidal representation

What is the relation between level and conductor of a supercuspidal representation of $\operatorname{GL}_2(\mathbb{Q}_p)$ for some prime $p$? Proposition 3.4 in Loeffler and Weinstein - On the ...
user15243's user avatar
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