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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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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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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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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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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A question on representation theory of p-adic groups
Let $V$ be a complex vector space of infinite dimension and let $(\pi,V)$ be a representation of the $p$-adic group $G:=GL_2(\mathbb{Q}_p)$. From representation theory, we know that if the ...
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Looking for concise books on automorphic L-functions, Eisenstein series on adelic homogeneous spaces
Indeed I am now trying to read a series of papers written by Einsiedler, Lindenstrauss, Michel and Venkatesh that study distribution of periodic torus orbits on homogeneous spaces. They make heavy ...
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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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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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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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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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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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Rankin-Selberg convolution and product of degrees
As I'm kinda obsessed with the Selberg class and because of the general converse conjecture, I'm still trying to get a rough idea of what automorphic representations and their L-functions as well as ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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Smallest Fourier coefficient divisible by a prime
If $f$ is a cusp form of weight $k$ and level $\Gamma_1(N)$. For simplicity, suppose that the Fourier coefficients of $f$ at $\infty$ are in $\mathbb{Z}$. Let $\ell$ be a prime which does not divide ...
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The notion of smoothness in the local situation
I am reading Bump's book on Automorphic forms and Representations and I am able to draw a lot of parallels between the theory of $GL(2, \mathbb{R})$ which is the infinite place and the theory of $GL(2,...
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Behaviour of a certain $L$ function at $s=1$
I was going through this paper. Corollary 7.3.4 says the $L$-function $L(s,\pi, \rm{sym}^4)$ is holomorphic except possibly at $s=0,1$ and gives a necessary and sufficient condition for it to have a ...
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A question on standard parabolic subgroup
Let $G$ be a connected reductive group over a number field $F$ and $P_0$ its minimal parabolic subgroup. Then we call a parabolic subgroup $P$ of $G$ is standard if $P_0 \subset P$.
Let $K$ be a ...
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The relation of Jacquet module and parabolic induction
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}
$Let $F$ be local field of characteristic zero and $(W,\langle,\rangle)$ be a $2n$-dimensional symplectic space over $F$.
Let $X,X^*$ be maximal ...
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Watson's triple product for automorphic forms shifted by Maass rising operators
Let $\phi_i$ be a holomorphic Hecke eigencusp form of weight $k_i$ for $\Gamma = \mathrm{SL}_2(\mathbb{Z})$, or a Maass cusp form (we then say that $k_i=0$). We assume they are normalised such that $\...
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Distance function on generalized upper half planes
Let $\mathbb{H}^n$ be the quotient $GL_n(\mathbb{R})/O(n) \cdot \mathbb{R}^\times$, which at least in the theory of automorphic forms is called a generalized upper half plane. We could also think of $...
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Questions on norms on Adelic group
This might be stupid question to some experts who works in the realm of automorphic form.
Let $K$ be a number field and $\mathbb{A}$ is a adele ring of $K$. Let $G$ be a connected reductive group ...
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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 ...
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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 \...
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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}_{\...
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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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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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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 \...
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Is an automorphic form of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ determined by its L-function?
To an automorphic representation $\pi$ of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ one can associate its L-function $s\mapsto L_{\pi}(s)$.
Is the map $\pi\mapsto L_{\pi}$ bijective?
Edit March ...
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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 ...
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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 ...
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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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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 ...
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What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$-adic field?
Let me stress that I am only interested in $p$-adic fields in this question, for reasons that will become clear later. Let me also stress that in some sense I am basically assuming that the reader ...
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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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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 ...
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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&...
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