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 ...

**3**

votes

**0**answers

85 views

### $p$-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}...

**8**

votes

**1**answer

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 ...

**0**

votes

**0**answers

62 views

### 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 ...

**1**

vote

**0**answers

48 views

### 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,...

**1**

vote

**0**answers

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 ...

**7**

votes

**1**answer

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) =...

**5**

votes

**2**answers

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 $G$ ...

**7**

votes

**4**answers

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, ...

**3**

votes

**1**answer

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.
...

**2**

votes

**1**answer

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}{...

**3**

votes

**1**answer

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}...

**2**

votes

**0**answers

78 views

### 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 ...

**4**

votes

**1**answer

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 $\...

**13**

votes

**0**answers

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 ...

**5**

votes

**0**answers

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} \...

**3**

votes

**0**answers

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).
...

**1**

vote

**0**answers

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 ...

**8**

votes

**0**answers

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},\...

**2**

votes

**0**answers

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 ...

**3**

votes

**0**answers

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 $M=GL_{n_1}(F) \times GL_{n_2}(F)$ ...

**9**

votes

**0**answers

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 $1 \le i \le m$, ...

**9**

votes

**2**answers

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 ...

**9**

votes

**0**answers

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 ...

**3**

votes

**0**answers

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.
...

**1**

vote

**0**answers

54 views

### 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}...

**8**

votes

**1**answer

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 $\...

**14**

votes

**2**answers

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 ...

**0**

votes

**0**answers

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^...

**18**

votes

**1**answer

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 (...

**4**

votes

**1**answer

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 ...

**10**

votes

**1**answer

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 = ...

**4**

votes

**1**answer

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 ...

**3**

votes

**2**answers

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 ...

**6**

votes

**0**answers

202 views

### 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 ...

**7**

votes

**0**answers

231 views

### 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 ...

**5**

votes

**0**answers

121 views

### 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}{\...

**3**

votes

**1**answer

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/...

**5**

votes

**1**answer

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 ...

**3**

votes

**0**answers

93 views

### 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 ...

**2**

votes

**0**answers

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 $PGL(2, F)$ ...

**4**

votes

**1**answer

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)...

**1**

vote

**0**answers

95 views

### 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 $D$, ...

**4**

votes

**1**answer

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 ...

**6**

votes

**1**answer

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 $L(s, Sym^k\Delta)$...

**5**

votes

**1**answer

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 ...

**4**

votes

**0**answers

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(...

**3**

votes

**0**answers

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 ...

**5**

votes

**0**answers

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 ...

**4**

votes

**0**answers

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})...

**5**

votes

**0**answers

74 views

### 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 ...