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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Does the Ramanujan-Petersson condition correspond to a Fourier type property?

The Ramanujan-Petersson is one of the requirements used in Selberg's class of L-functions, and as such is a necessary condition for the Riemann Hypothesis to hold. The general converse conjecture ...
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3 votes
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300 views

Supercuspidal, spherical and discrete series representation

Let $G$ be an algebraic group defined over $\mathbb{Q}$ with maximal unipotent radical $N$. Let $\pi$ be an admissible representation of $G(\mathbb{Q}_p)$, we say that this representation is ...
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The link between Satake parameter and Godement-Jacquet L-function of an automorphic representation of $GL_{n}$

Origin of the question: I'm reading the following survey of K. Martin, more generally I'm looking for the "best way" to define L-function associated to an automorphic representation of a ...
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Does any element of an L-rig fulfill Rudnick and Sarnak hypothesis H and thus Selberg orthogonality conjecture?

I introduce the notion of L-rig in the first paragraph of Are there infinitely many L-rigs? Calling "genuine L-function" any element of an L-rig, and to the light of recent results about L-...
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8 votes
1 answer
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Adelization of automorphic forms for higher class number

Short version: is there a canonical way to adelize a classical Hecke eigenform automorphic form when the adelic quotient has many components? If not, what are the different "choices", how ...
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Explicit Satake isomorphism

Consider the group $G=GL(n)$ over a non-Archimedean field and $K$ a maximal compact subgroup. Let $(\pi, V)$ be a smooth admissible irreducible representation of $G$, say spherical. I would like to ...
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Explanation about Lapid-Rallis iductive argument (doubling method)

I am reading Lapid-Rallis "On the local factors of representations of classical groups" and I am completely stuck with the proof of Proposition 3. In the case $\mathcal V$ is not anisotropic,...
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Classical and adelic automorphic forms from SL(n) to GL(n) over number fields

It's a long post but I felt like I needed to provide some context to my problem. The explicit questions start at the bold font questions below. In the classical world, it seems that one is usually ...
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4 votes
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On the precise form of $\mathrm{GL}(3)$ (and others) L-functions

$\DeclareMathOperator\GL{GL}$Typical $\GL(3)$ automorphic L-functions are often depicted as looking like the following Dirichlet series: $$L(s, \pi) = \sum_{n >0} \frac{A(1,n)}{n^s}$$ Is there a ...
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The identity connected component of centralizers of unipotent orbits

This is, in a way, a follow up question to Unipotent orbits and intersection with Levi and pseudo-Levi subgroups. I was reading "A generalisation of the Bala–Carter theorem for nilpotent orbits&...
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Symmetric square L-function with non square-free level

Let $f$ be a primitive holomorphic cusp form of weight $k$, level $N$ and nebentypus $\chi$, with its $L$-function $L(s,f)=\displaystyle\sum_{n\geq1}\lambda_f(n)n^{-s}$ for $\mathrm{Re}(s)>1$. Let $...
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Two basic questions on congruence subgroups

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$I have two questions related to congruence subgroups. Let $$\Gamma=\Gamma_0(N)=\Big\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \...
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Whether or not the Maass form for $\Gamma _0(N)$ on $GL(3)$ covers the classical symmetric lift of a newform on $GL(2)$?

I have a blur which needs some help from the experts here, and may look naive for some experts. Recently I read Zhou's paper "The Voronoi formula on $GL{(3)}$ with ramification" (https://...
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Unipotent orbits and intersection with Levi and pseudo-Levi subgroups

Given a simple complex Lie group $G$ (I might say upfront that I am mostly interested with exceptional Lie algebras) and a nilpotent orbit $\mathcal{O}\subset G$ I would like to describe the ...
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4 votes
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From $\mathrm{GL}_{1}/K$ to $\mathrm{GL}_{2}/\mathbb{Q}$, where $K$ is a cyclic cubic extension

(Migrated from MSE) Let $K$ be a cyclic cubic extension of $\mathbb{Q}$. For example, one can take simplest cubic fields. By automorphic induction (which is known for cyclic extension of prime degree ...
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Converse of Gelbart-Jacquet lift for $\mathrm{GL}(3)$ Maass forms

(This question is migrated from MSE) Gelbart-Jacquet lift gives functoriality from $\mathrm{GL}(2)$ to $\mathrm{GL}(3)$ that corresponds to a symmetric square map $\mathrm{Sym}^{2}: \mathrm{GL}(2, \...
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3 votes
1 answer
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How to construct a group of Möbius transformations corresponding to a given fundamental triangle?

Most introductory textbooks on the modular group begin with an introduction of it as the group generated by the two Möbius transformations: \begin{gather*} z'=z+1 \\ z'=-\frac{1}{z} \end{gather*} and ...
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Hoffstein–Lockhart for non-congruence subgroups

Let $\Gamma$ be a non-congruence subgroup of $\operatorname{SL}(2,\mathbb{Z})$ of finite index and let $f$ be a holomorphic cuspidal modular form of weight $k$ for the group $\Gamma$. For simplicity, ...
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4 votes
1 answer
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$\DeclareMathOperator\SL{SL}$Multiplicities of irreducible representations in discrete part of $L^2(\SL(2,\mathbb{Z})\backslash{\SL(2,\mathbb R)})$

$\DeclareMathOperator\SL{SL}$It is well-known that the cuspidal (or discrete) part of $L^2(\SL(2,\mathbb{Z})\backslash{\SL(2,\mathbb{R})})$ decomposes into irreducible representations of $\SL(2,\...
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Global Waldspurger packet is finite or infinite?

$\DeclareMathOperator\Mp{Mp}$Let $F$ be a number field and $\pi$ be an irreducible cuspidal automorphic representation of $\operatorname{PGL}_2(\mathbb{A}_F)$. Then we can think a submodule $L_{\pi}^2$...
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6 votes
1 answer
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Global symplectic (orthogonal) type of automorphic representation compels its type to all its local components?

Let $F$ be a number field. For an irreducible cuspidal automorphic representation $\pi$ of $\operatorname{GL}_n(\mathbb{A}_F)$, we say that $\pi$ is symplectic (or orthogonal) if $L(s,\pi,\bigwedge^{2}...
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How to check the non-emptyness of the A-packet of non-split $\operatorname{SO}(2n+1)$?

Let $F$ be a number field and $G_n=\operatorname{SO}(2n+1)$ be the split group over $F$. Let $G_n^{\times}$ be a non-split group over $F$. Let $\tau$ be an irreducible cuspidal automorphic ...
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3 votes
1 answer
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Why differential operator preserves $K$-finiteness of automorphic form?

Let $G$ be a reductive group over a number field $\mathbb{Q}$ and $K$ be a maximal compact subgroup of $G$. Let $\Gamma$ be an arithmetic subgroup of $G(\mathbb{Q})$. Let $\mathfrak{g}$ be the ...
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9 votes
2 answers
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Computing the Petersson norm of newforms of weight 2 from the symmetric square $L$-function

Let $f \in S_2(\Gamma_0(N))$ be a newform with trivial character. I want to compute the Petersson norm $\lVert f\rVert^2$ of $f$, not normalized by $1/[\operatorname{SL}_2(\mathbf{Z}):\Gamma_0(N)]$, ...
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2 votes
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Local A-packet is singleton for unramified place?

Let $\pi$ be a generic $A$-parameter, that is an isobaric automorphic representation of linear group. Decompose $\pi= \otimes \pi_v$ as a restricted tensor product. Then by the local Langlands ...
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Generic Arthur-parameter of symplectic group

For a irreducible cuspidal automorphic representation $\pi$ of $Sp(2n)$, we can attach its generic $A$-parameter, that is isobaric sum automorphic representation of $GL_{2n+1}$. I know it is of the ...
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Does Rankin-Selberg convolution preserve primitivity?

Call $L$-function any element of an L-rig (see Are there infinitely many L-rigs? for a definition). Suppose $F$ and $G$ are two primitive L-functions. Is $F\otimes G$ itself primitive? If yes, does ...
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Is it possible $L(\frac{1}{2},\phi \times \phi')=0$ for all $\phi'$?

Let $\phi$ be an irreducible cuspidal automorphic representation of $GL_n(\mathbb{A})$ of symplectic type, that is, the exterior square $L$-function $L(s,\phi,\Lambda^2)$ has a pole at $s=1$. Then I ...
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4 votes
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Generalizations of the idea of automorphic

The notion of an automorphic form/representation (and sometimes, of Langlands program in tandem) has been extended in many directions - from arithmetic to geometric to topological - but two versions I ...
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Global Vogan A-packet is infinite set?

For an cuspidal automorphic representation of general linear group, we can attach its global Vogan A-packet. Though I thought that it is finite set, in some paper, it is written that there are ...
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3 votes
1 answer
293 views

Tannakian fundamental group of automorphic representations

Let $\mathcal{C}_{\mathrm{aut}}(G, F)$ be the category of automorphic representations of a connected reductive group $G$ over a number field $F$. If this is a Tannakian category, it has an associated ...
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4 votes
1 answer
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Schur lemma and Whittaker functions

$\DeclareMathOperator\GL{GL}$Let $(\pi,V)$ be an infinite dimensional irreducible admissible representation of $\GL_2(\mathbb{Q}_p)$. Let us fix an element $v_0\in V$ and define a vector space $$V_{...
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7 votes
1 answer
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On a certain double integral appearing in the Fourier series coefficients of $\mathrm{SL}_2(\mathbb{C})$-Eisenstein series

The following integral appears naturally within the computation of the Fourier series coefficients of a real analytic $\mathrm{SL}_2(\mathbb{C})$-Eisenstein series: \begin{align*} \int_{-\infty}^{\...
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1 vote
0 answers
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Global tempered A-parameter is the same with generic A-parameter?

It seems there are two relevant terminologies on global Arthur parameter. One is generic A-parameter and the other one is tempered A-parameter. I thought that these two terminology are same. But is ...
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2 votes
1 answer
106 views

Non-vanishing criterion of the Hom space of induced representation of p-adic groups?

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\GL{GL} $Let $F$ be local field of characteristic zero and $(W,\langle,\rangle)$ be a $2n$-dimensional symplectic space over $...
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9 votes
1 answer
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A question on the period integral of Rankin-Selberg $L$-function

$\DeclareMathOperator\GL{GL}$Let $\Pi$ and $\pi$ be irreducible automorphic representations of $\GL_{n+1}(\mathbb{A}_F)$ and $\GL_n(\mathbb{A}_F)$ respectively, where $n \geq 2$, $F$ is a number field ...
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2 votes
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Unramified constituent of Weil representation of $U(2)$

Let $E/F$ be a quadratic extension of local field of characteristic zero. Let $\omega$ be the quadratic character of $F^{\times}$ associated to $E/F$ by local class field theory and $\gamma:E^{\times} ...
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1 vote
1 answer
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Part of some generic representation is also generic?

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\GL{GL} $Let $F$ be local field of characteristic zero and $(W,\langle,\rangle)$ be a $2n$-dimensional symplectic space over $...
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2 votes
0 answers
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Automorphic representations for two number fields

I have a basic question that I have not found a reference for: how are the automorphic representations of a reductive group $G$ for two different number fields $K \supset K'$ related? For example, if $...
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Constant coefficient of Eisenstein series

Let $\chi$ be a character of $\mathbb{Q}^{\times}\setminus\mathbb{A}^{\times}$. We define an induced representation of $Mp_2(\mathbb{A})\simeq SL_2(\mathbb{A})\times \mathbb{C}^1$, $$I(s,\chi) := \{\...
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2 votes
1 answer
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Galois representations attached to a cusp form for different primes

If I have a cusp form $f$, I can consider the associated Galois representation $\rho_l(f)$ for any prime $l$. For two distinct primes $p$ and $q$, what is the relationship between $\rho_p(f)$ and $\...
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4 votes
1 answer
252 views

Adelization for any classical arithmetic subgroup

In the classical setting, we can define automorphic forms on $\text{SL}_n(\mathbb{R})$ with respect to any lattice $\Gamma$. In fact, for $n \geq 3$, all lattices are arithmetic subgroups. I have ...
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2 votes
0 answers
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Maass forms of higher weight for GL(3)

In the case of $GL(2)$ there is a notion of Maass form of weight $k$, precisely they are eigenfunctions of the weight $k$ Laplacian operator, $\Delta_k$ (taken from "Automorphic forms and ...
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4 votes
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Values at 1 of symmetric power L-functions of Maass cusp forms

I have a blur that whether one has $L(1,\text{sym}^2f)\ll \log^A q$ for some $A>0$? Here $f$ is assumed to be a Maass cusp form of square-free level $q$. If any experts here know something about ...
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What can the $p$-component of an automorphic representation be once we have fixed its $\infty$-component?

My question concerns Fargues' 2004 Astérique paper "Cohomologie des espaces de modules de groupes $p$-divisibles", available here. I will use the same notations below. Before stating it, I ...
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0 votes
0 answers
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The specific connection between the Hecke operator and the t'Hooft Operator

As I was reading some articles concern about the Selberg trace formula and its general form, I have noticed that the Selberg trace formula and its general form can be understand as the energy spectrum ...
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1 vote
1 answer
230 views

How to relate Rankin triple L-function to its Dirichlet series

I have a very tricky question which may look naive to many experts here. Let $f$ be a newform of level prime $P$, and $g,h$ two newforms of level 1, respectively. These three forms $f,g,h$ are all of ...
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1 vote
0 answers
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The pole of symmetric square $L$-function of $GL(n)$ at $s=1$

Let $\pi$ be an irreducible cuspidal automorphic representation of $GL(n)$. Suppose the symmetric square $L$-function of $\pi$ $L(s,\pi,Sym^2)$ has a pole at $s=1$. Then since $L(s,\pi \times \pi)=L(s,...
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5 votes
2 answers
222 views

A lower-bound for the square-mean of Fourier coefficients of cusp forms at primes argument

There is a basis question which puzzles me for a while. The question is the following: Let $X\ge 2,$ and $\lambda(n)$ be the $n$-th Fourier coefficient of a $GL(2)$ newform of prime level $N>1$, ...
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  • 151
2 votes
1 answer
220 views

Jacquet module and Frobenius reciprocity

Let $F$ be a local field of characteristic zero and $G$ be a classical group over $F$. Let $P=MN$ be a parabolic subgroup of $G$ and $\pi$ a irreducible smooth representation of $M$. Let $\sigma$ be ...
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