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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 ...

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$p$-adic spectral problem?

Warning: this is a very very naive question. Spectral problem is one of the important subjects in the theory of automorphic forms (as I believe). If $\Gamma$ is a discrete subgroup of $\mathrm{SL}_{2}...
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1answer
117 views

Finite multiplicities

Let $G$ be a locally compact group and $\Gamma$ a lattice in $G$. Is it known whether the space $$ \mathrm{Hom}_G\left(\pi,L^2(\Gamma\backslash G)\right) $$ is finite dimensional for $\pi\in\widehat ...
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Is a tempered representation globally generic?

I know there is an general belief that globally generic representation is tempered. I am wondering whether the converse is known, that is tempered representation is generic? If it is not known, is ...
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Does Hom functor preserve restricted tensor product?

Let $\pi$ is an automorphic representation of reductive group $G$. Then we can decompose $\pi = \otimes \pi_v$ as restricted tensor product by Flath theorem. I am wondering whether $\text{Hom}_G(\pi,...
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33 views

Mean value estimates for general number fields

Results are known in many different cases to bound powers of L-functions on average over a wide enough family. I am interested in results for general number fields, not only for the rationals, for ...
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1answer
276 views

Relation between Fourier coefficients and Satake parameters

Let $L(s)$ be an automorphic L-function (attached to a self contragredient automorphic representation on $GL(3)$), according to the following notations for $s$ of sufficiently large real part: $$L(s) =...
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2answers
170 views

Definition of unitary representation of $\mathbf G(\mathbb A_k)$

Let $k$ be a global field, and let $G = \mathbf G(\mathbb A_k)$ for a connected, reductive group $\mathbf G$ over $k$. In these notes by Jayce Getz and Heekyoung Hahn, a unitary representation of $G$ ...
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4answers
339 views

Fourier expansion at inequivalent cusps

Let $\Gamma\subset SL(2,\mathbb{R})$ be a Fuchsian group of the first kind. Let $c_1, c_2$ be inequivalent cusps of $\Gamma.$ Consider $f\in M_k(\Gamma)$ a weight $k$ holomorphic automorphic form, ...
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1answer
142 views

Eisenstein series for discrete subgroups of SL(2,C)?

I am looking for a reference for Eisenstein series for discrete subgroups of $SL(2,\mathbb C)$, in particular, finite index subgroups of $SL(2,\mathcal O_K)$ where $K$ is an imaginary quadratic field. ...
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1answer
101 views

$L^2(X) \cong L^2(X',\xi)$

Recently, I read a notes about Sakellaridis and Venkatesh conjecture. It mentions a technique called "unfolding" and gives an example: Let X=A\G, X'=N\G, where G=PGL(2), A={ $\left[\begin{array}{...
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1answer
117 views

Hecke algebra $\mathcal H(\operatorname{GL}_2(\mathbb Q_p)/\operatorname{GL}_2(\mathbb Z_p))$ and Hecke operators

I was reading James Cogdell's notes here on automorphic representations and came to the following claim about the spherical Hecke algebra $\mathcal H(\operatorname{GL}_2(\mathbb Q_p), \operatorname{GL}...
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Definition of Base Change of representation

Recently, I read the paper on Ichino-Ikeda conjecture for unitary groups by Raphaël Beuzart-Plessis (https://arxiv.org/pdf/1602.06538.pdf). In the introduction, it says that Let $BC(\Pi)$ be the ...
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1answer
162 views

Local L-function $L(s,\pi_p\times \chi_p)=1$

Let $\pi_p$ be a ramified representation of $GL(n,\mathbb{Q}_p)$. Let $\chi_p$ be a ramified representation of $GL(1,\mathbb{Q}_p)$. Is it generally known that $L(s,\pi_p\times \chi_p)=1$ if $\...
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316 views

Lifting automorphic Galois representations to arithmetic fundamental groups and their quotients

Suppose $V$ is an algebraic variety over a number field $K$. The absolute Galois group $G_K$ of $K$ acts by outer automorphisms on the étale fundamental group $\pi_1(V_{\bar{K}})$ where $\bar{K}$ is ...
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96 views

Integrals of modular forms

Setup: Write $G = \text{SL}_2(\mathbf{R})$ and $\Gamma = \text{SL}_2(\mathbf{Z})$. Let $f$ be a modular form on $\mathbf{H}$ of weight $2k$, so that $$f(gz) = f(z) (cz + d)^{2k} \qquad \text{for} \...
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168 views

Fricke involution on GL(3)

Define $\Gamma_0(N)=\{\begin{pmatrix} a&b&c\\ d&e&f\\ g&h&i \end{pmatrix} \in SL(3,\mathbb{Z})|g\equiv h\equiv 0(\mod N)\}$ be the $N$-level congruence subgroup on GL(3). ...
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93 views

Some simple question of the base change of the unitary group to general linear group

Let $E/F$ be a quadratic extension of number fields and $\chi$ is a unitary automorphic character of $E^{\times}$. Let $\pi$ be an automorphic representation of $U(n)(F)$ associated to $E/F$, which ...
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155 views

Surjectivity of reduction for Hilbert modular forms

Fix a totally real field $K$, a level $\mathfrak{n}$, a (parallel) weight $k\geq 2$ and a primitive ray class character $\chi$ modulo $\mathfrak{n}$. Then one can form the space $S_k(\mathfrak{n},\...
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125 views

Two basic question on parabolic induction

I want to ask some basic two questions on the parabolic induction. Let $F$ be a local fields. Let $\chi_1,\chi_2$ be two characters of $GL_1(F)$ and $GL_1 \times GL_1$ be the Levi part of the ...
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101 views

Some basic question on the parabolic induction

I would like to ask some basic question about parabolic induction. Let $F$ be a local field and $G=GL_n(F)$ and $P=MN$ its parabolic subgroup whose Levis subgroup $M=GL_{n_1}(F) \times GL_{n_2}(F)$ ...
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182 views

Some question on the induced representation of tensor product

I would like to some question concering the induced representation of tensor product. Let $F$ be a local fields of charateristic 0 and $0<a<m$ are two positive integers. For $1 \le i \le m$, ...
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2answers
622 views

Kuznetsov trace formula, orthogonality of Bessel functions

Sorry if this is a vague question. I remember from my younger days that before proving his trace formula, Kuznetsov had a pretty result on orthogonality of Bessel functions. The formulas that I am ...
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217 views

The trace formula over function fields

There are many examples in number theory where an "arithmetic" problem (i.e. for number fields) has an easier analogue for function fields over finite fields. This is also true for questions ...
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65 views

Archimedean components of base changed automorphic representations

Let $\pi$ be an automorphic (cuspidal) representation of ${\rm GL}(n)_{/\Bbb Q}$. Let $K\supset\Bbb Q$ a numberfield. The existence of the base change lift $\pi_K$ is known in a number of cases. ...
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The Geometry of Jacobi Forms and their Asymptotic Expansions

A Jacobi form of weight $k$ and index $m$ is a meromorphic function $\varphi: \mathbb{H} \times \mathbb{C} \to \mathbb{C}$ satisfying $$\varphi\bigg(\frac{a \tau + b}{c \tau + d}, \frac{z}{c \tau + d}...
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1answer
272 views

Definitions of $\pi_1 \times \pi_2, \pi_1 \boxplus \pi_2, \pi_1 \boxtimes \pi_2$

Let $\pi_i$ be a smooth, admissible (possibly irreducible) representation of $\operatorname{GL}_{n_i}(k)$ for $k$ a $p$-adic field. I have seen the following representations defined in terms of $\...
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2answers
751 views

What is the archimedean Hecke algebra?

Let $\mathbf G$ be a connected, reductive group over $\mathbb Q$. For each nonarchimedean place $v$, let $K_v$ be a maximal compact subgroup of $\mathbf G(\mathbb Q_v)$. The space $\mathscr H(\mathbf ...
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53 views

Generators of ideals in the quadratic extensions

Let $M = K[[X,Y,T]]$ and $R = K[[X,Y]]$ be a power series ring. For elements $a_2, b_2, ... , a_n, b_n$ belonging to $R$, let us define the ideal $I$ of $M$ with $n$ number of generators as $$I :=(T^...
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583 views

What computer program for automorphic forms

This question has its origins in this entertaining discussion on MO. There are many programs (CAS) and libraries that are able to handle algebraic expressions. These are both a verification tool for (...
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1answer
168 views

Meromorphic continuation of local zeta integrals

Let $f$ be a Maass cusp form for $\text{SL}_2(\mathbb{Z})$ on the upper half plane. Let $\varphi_0$ be its lift to an automorphic form on $G = \text{PGL}_2(\mathbb{R})$ and let $\pi = \pi_{f} =\langle ...
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1answer
466 views

Estimating volumes in the trace formula

Let $G$ be a reductive group. In many instances of the trace formula, elliptic terms corresponding to the $G(k)$-conjugacy class of $\gamma \in G(k)$ are weighted by the following volumes $$v_\gamma = ...
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1answer
79 views

Integrality of the support of matrix coefficients?

Consider a division quaternion algebra $D$ over a number field $F$. For an automorphic representation $\pi$ of $D$, I am interested in the associated matrix coefficients $$f : \gamma \in G \longmapsto ...
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122 views

Square-integrability of non-holomorphic Poincare series

In what follows, $\mathfrak{H}$ denotes the upper half plane, and $\Gamma=SL(2,\mathbb{Z})$. On the modular curve $\Gamma\backslash\mathfrak{H}$, consider the non-holomorphic Poincare series, defined ...
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Galois action on functions on an adelic coset space

For a reductive group $G$ over a number field $K$, an automorphic representation for $(G, K)$ is an irreducible admissible constituent of the right regular representation of $G(\mathbb{A}_K)$ in the ...
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Galois representations associated to the modular tower and automorphy

Consider $\mathcal{M} = \{M_{g,n}, \mu_{g,n}^{g’,n’}\}$, the system of moduli spaces of $n$-pointed smooth algebraic curves along with the basic maps amongst them coming from identifying and ...
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Direct sum decomposition of the space of cuspidal automorphic forms

$\newcommand{\G}{\mathbb{G}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\A}{\mathbb{A}} \newcommand{\Autom}{\mathcal{A}} \newcommand{\cen}{\mathcal{Z}} \newcommand{\lieg}{\...
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235 views

Modular forms, Maass forms and Automorphic representations

I am beginning to learn about automorphic forms, and stay perplex concerning the two languages of "forms" versus "representations" often used at the same time. As far as I understand, a modular/...
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1answer
101 views

Fields of rationality as a notion of automorphic size

I want to interpret the degree of the field of rationality of an automorphic form as a notion of size, analogously to the conductor, and this question is about the possible obstructions to do so. The ...
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For Hida theory on $GU(2,2)$ can $p$ be inert in the imaginary quadratic field $K$?

I am familiar with the theory of Hida families of modular forms, so Hida theory on $GL_2$, but I am not familiar with Hida theory on any other algebraic groups. My question concerns Hida families of ...
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66 views

Continuity of the conductor of automorphic representations

I am interested in properties of (semi-)continuity of the conductor of automorphic representation. Let $F$ be a number field. The meta question is, given a function on the unitary dual of $PGL(2, F)$ ...
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1answer
152 views

Are all these representations supercuspidal

Let $D$ a division quaternion algebra over a number field $F$, and consider $(V,q)$ be a $D$-hermitian space of $D$-dimension $2$, and introduce its group of isometries \begin{align*} \mathrm{GU}(V, q)...
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Is there an analysis theorem analogous to Kuznetsov/Petersson trace formula?

I am thinking about general differential operator acts on a compact manifold. Is there something similar to Kuznetsov trace formula? For example, let $f_i $ be the eigenfunctions of an operator $D$, ...
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1answer
174 views

Making sense out of intertwining operators defined by a vector valued integral

Let $G$ be the rational points of a connected, reductive group over a $p$-adic field $F$. Let $S$ be a maximal split torus of $G$ with $\Delta$ a set of simple roots corresponding to a minimal ...
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1answer
215 views

Symmetric powers of Ramanujan tau-function

Let $\Delta(z)$ be the modular form associated with Ramanujan $\tau$-function. For any $k=2,3,...$, $Sym^k\Delta$ is conjectured to be an automorphic form on $\mathrm{GL}(k+1)$ and $L(s, Sym^k\Delta)$...
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1answer
259 views

Haar measure on $\mathrm{SL}_3(\mathbb{Z}) \backslash \mathrm{SL}_3(\mathbb{R}) / \mathrm{SO}_3(\mathbb{R})$

The bi-invariant Haar measure on the quotient $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathrm{SL}_2(\mathbb{R}) / \mathrm{SO}_2(\mathbb{R})$ represents the moduli space of rank two real lattices modulo ...
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96 views

Dimension of space of K-fixed vectors

If $G$ is an unramified group over an $p$-adic field $F$, the Satake isomorphism identifies the spherical Hecke algebra with respect to a special maximal compact subgroup $K$. In particular, (1) $H(G(...
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120 views

Equivalence of formulations of Ihara's lemma

I'm wondering about the relationship between two formulations of Ihara's lemma for $\text{GL}_2$ I've seen: (1) the "concrete" version given in, for example, Darmon, Diamond, and Taylor, which says ...
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114 views

What is a Hilbertian stack?

In describing spectral decompositions of some $L^2$ spaces, Moeglin-Waldspurger (Spectral Decomposition and Eisenstein Series) keep using the term "Hilbertian stack" (e.g. the Corollary in V.3.14) but ...
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108 views

Orbits of arithmetic subgroups intersection a compact set

Let us suppose we have $G$ a connected reductive group over a number field $F$. Consider $G(\mathbb{A})$ the group over the adeles and $G(\mathbb{Q})$ embedded discretely. For $\gamma \in G(\mathbb{Q})...
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Example of infinite automorphic multiplicity

Let $G$ be a locally compact group and $\Gamma$ a lattice in $G$. For an irreducible unitary representation $\pi$ of $G$ let $$ m_\Gamma(\pi)=\dim\mathrm{Hom}_G(\pi,L^2(\Gamma\backslash G)) $$ be its ...