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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Question on the genericity of unramified representation

Let $F$ be a $p$-adic local field and $W$ be a 2n-dimensional symplectic space over $F$. Let $G_n$ be the isometry group of $W$ and $B_n$ be the Borel subgroup of $G_n$. Then the maximal torus $T_n$ ...
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What is the "weight" of an automorphic form for $\mathrm{PGL}_2$?

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PGL{PGL}$I'm trying to understand what the notion of "weight" is for automorphic forms over $\GL_2(F)$ where $F$ is some number field, in ...
HASouza's user avatar
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Reference Request: Possible generalizations of the stability of $\gamma$-factors

$\DeclareMathOperator\GL{GL}$ Let $F$ be a nonarchimedean local field. Suppose $\pi, \sigma$ are irreducible admissible representations of $\GL_{n}(F)$ and $\GL_{m}(F)$ respectively, with $n \geq m$. ...
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Question on generic A-packet

Let $G$ be a classical group and $\phi$ be a generic $A$-parameter of $G$. I am wondering whether each automorphic representations in the $A$-packet associated to $\phi$ are locally generic at almost ...
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Simple question on the genericity of induced representation

$\DeclareMathOperator\GL{GL} \DeclareMathOperator\Sp{Sp} \DeclareMathOperator\Ind{Ind}$ Let $F$ be a $p$-adic field and $\Sp(2n)$ symplectic group over 2n dimensional symplectic space over $F$. Let $B=...
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Question on the unramified representation

$\DeclareMathOperator\GL{GL}$Let $F$ be a $p$-adic field and $\chi$ be an unramified character of $\GL_1(F)$. Consider an induced representation $\pi$ of $\GL_2(F)$ induced from the character $\chi|\...
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Symmetric power lift of modular forms

Let $f_1$ and $f_2$ be two cuspforms of weights $k_1$ and $k_2$ and nebentypus $\epsilon_1$ and $\epsilon_2$ respectively such that $f_1 \neq f_2 \otimes \chi$ for some Dirichlet character $\chi$ of ...
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Iwahori action on the $p$-ordinary line of a principal series representation

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\diag{diag}\DeclareMathOperator\Ind{Ind}\newcommand\Iw{\mathrm{Iw}}\DeclareMathOperator\ord{ord}$Let $F$ be a $p$-adic local field, i.e. a finite ...
Hetong Xu's user avatar
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How to see that Eisenstein series are eigenfunctions of the laplacian?

Let $\Gamma$ be a discrete subgroup of $PSL_2(\mathbb{R})$ of finite type. Let $c_1,\ldots,c_h\in\mathbb{R}\cup\{\infty\}$ be a set of representatives of the $\Gamma$-equivalence classes of cusps. For ...
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Is there any notion of Poincaré series for Hermitian modular forms?

I have been studying modular forms and their generalisations for a year or so. It is a very interesting fact that the space of cusp forms $S_k$ is generated by the Poincaré series of exponential type (...
Ancient Antagonist's user avatar
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Reference Request: Local decomposition of GGP period integrals of cuspidal forms on unitary groups

Setup: Let $E/F$ be a CM-extension of global number fields. Let $(V,\phi)$ be an Hermitian space of dimension $n$ over $E$. Let $(V^{\flat}, \phi^{\flat})$ be a subspace of $V$ of dimension $n-1$ on ...
Hetong Xu's user avatar
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Analyticity of unramifed part of Rankin-Selberg $L$-functions on $\Re(s)=1$

I have only a little knowledge about automorphic representations and $L$-functions. Now I am reading the textbook of Goldfeld and Hundley on automorphic representations, and also planning to read the ...
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$p$th Fourier coefficients of newforms for ramified primes $p$

This question is about some basic(classical) results on Atkin-Lehner-Li theory of newforms. Let $f$ be a (normalized) newform of level $N$ and character $\epsilon$. Denote the $n$th Fourier ...
LWW's user avatar
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'$\times$' or '$\otimes$' when writing $L$-functions?

Recently, I came across the Langlands correspondence theorem, there is the following line: $$L(s,\pi(\sigma) \times \pi(\tau)) = L(s,\sigma \otimes \tau), $$ where $\sigma$ and $\tau$ are ...
Misaka 16559's user avatar
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A question on the averages of Kloosterman sums

Sorry to disturb. Recently, I encountered a puzzle on the sums involving two Kloosterman sums. That is, For any $h, q_1,q_2\in \mathbb{N}$ with $(q_1,q_2)=1$ and $Q>1$, how two get a bound $$\sum_{...
hofnumber's user avatar
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Automorphic representation of GL(1)

These might be very silly questions, but somehow I am not able to understand it or I might have misunderstood something. I am reading automorphic forms from this book. What I have understood till now: ...
user15243's user avatar
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On the square mean of Fourier coefficients of cusp forms

I have a question which may look naive for many experts here: For any primitive holomorphic form $f$ of level $M$ ($M\in \mathbb{N}$), whether or not one has the lower bound that: $$\sum_{X<n\le 2X}...
hofnumber's user avatar
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Explicit description for action of Weyl element in Whittaker model for GL2

Let $F$ be a non-archimedean local field and let $\pi =\mathscr{B}(\chi, \chi^{-1})$ be a principal series representation of $\mathrm{PGL}_2(F)$ induced from a character $\chi$ of $F^\times$. Let $w = ...
Steph Curry's user avatar
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Question of pole and zeros of symmetric or exterior global $L$-function of $\mathrm{GL}_n(\mathbb{A})$

Let $\pi$ be a unitary cuspidal representation of $\mathrm{GL}_n(\mathbb{A})$. It is written is some paper that using the results towards the generalized Ramanujan conjecture in the paper "On the ...
Andrew's user avatar
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Question on the Rankin-Selberg epsilon function

Let $\pi$ and $\pi'$ be unitary cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A})$ and $\mathrm{GL}_m(\mathbb{A})$, respectively. It is well known that the complete Rankin-Selberg $L$-...
Andrew's user avatar
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Modular forms and the cocycle condition in group cohomology

I am interested in $H¹$ right now and the cocycle condition $φ_{jk} • φ_{ij} = φ_{ik}$ because of how it is said to relate to automorphic forms. I can't quite see the relationship between factors of ...
Ronald J. Zallman's user avatar
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Consult a question about subconvexity bounds for symmetric-square L-functions in an Arxiv-eprint due to P. D. Nelson

Sorry to disturb, the experts here. Recently, I read a paper of Nelson ("Subconvex equidistribution of cusp forms: reduction to Eisenstein observables--"https://arxiv.org/pdf/1702.02908.pdf)....
hofnumber's user avatar
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Explicit construction of $T$-orbits of generic characters of unitary groups

Let $F$ be a $p$-adic field. Let $E$ be a quadratic extension of $F$ and $G$ be a quasi-split unitary group $U(2n)$ or $U(2n+1)$ over with respect to $E/F$. Let $N_{E/F}$ be a norm map. Let $B=TU$ be ...
Andrew's user avatar
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Can Langlands correpondence be restated using topos?

Langlands correspondence describes an equivalence between Galois representations and automorphic representations under some conditions. Laurent Lafforgue applying Olivia Caramello thesis described in ...
jaylooker's user avatar
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Question on automorphic $L$-functions

Let $\pi$ be an automorphic representation of $\textrm{GL}_n$. Associated to $\pi$, we can define the standard $L$-function $L(s, \pi)$. My question is: what is the difference between $L(s, \pi)$ and ...
dekimashita's user avatar
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The Wilton-type bounds involving half-integral weight cusp forms

There is a basic question which puzzles me for a while, and maybe look naive for some experts here. The question is the following: Let $f(z)=\sum_{n\ge 1} a_f(n) n^{k/2-1/4}e(nz)\in S_{k+1/2}(4N)$ be ...
hofnumber's user avatar
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Can any pair of associate parabolics be related by opposite parabolics?

Let $G$ be a reductive group, say over an algebraically closed field of characteristic zero. We have the following definitions for a pair of parabolic subgroups $P_1$ and $P_2$ with Levi quotients $...
Anthony Blanche's user avatar
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On the notion of cuspidality

Let $k/\mathbb{Q}$ be a number field and $\mathbb{A}$ its ring of adèles. As usual $\mathbb{A} = \mathbb{A_f} \times \mathbb{A_{\infty}}$. The standard definition of an automorphic representation $(\...
Maty Mangoo's user avatar
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Shintani's unpublished paper on automorphic forms

I'm trying to find Shintani's preprint: Shintani T., On automorphic forms on unitary groups of order 3, unpublished, 1979. It seems to be impossible to find, even though several authors quote it. I ...
Fra's user avatar
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Several L-functions but one Galois representation: How to choose

Let $\mathbf{G}$ be a reductive group which enjoys all the nice properties a reducive group can dream of. Fix $(\mathbf{G},X)$ a Shimura datum associated with it and assume that if $K\leq\mathbf{G} $ ...
Marsault Chabat's user avatar
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1 answer
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Experiments with Voronoï summation

In order to test my understanding of the Voronoï summation formula, I tried to apply it to a simple estimation of partial sums of Fourier coefficients of cusp forms. The result I obtained cannot ...
user50139's user avatar
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7 votes
1 answer
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Automorphic classification of different types of abelian surfaces

For elliptic curves over $\mathbb{Q}$ the Mumford-Tate group is either $\mathrm{GL}_2$ or $\mathrm{Res}_\mathbb{Q}^F (\mathbb{G}_m)$ if it has CM with the imaginary quadratic field $F$. In this case ...
Alireza Shavali's user avatar
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Discrete subgroups of $\text{Sp}_{4}(\mathbb{Q})$ parameterizing polarized Abelian surfaces plus torsion data

I want to start by considering a familiar congruence subgroup of the integral symplectic group $\text{Sp}_{4}(\mathbb{Z})$. For a positive integer $N$, let $\Gamma _{0}^{(2)}(N) \subset \text{Sp} _{4}(...
Benighted's user avatar
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Question on the relation of global theta lifting and local theta lift

Let $F$ be a number field and $G$ (resp. $H$) an odd orthogonal (resp. metaplectic group) over $F$. Let $v$ be a finite place of $F$ and $\sigma_v$ a supercuspidal representation of $G_v(F_v)$. Let $\...
Andrew's user avatar
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Correspondence between motives and automorphic representations

What I know: I understand motives via its realization; in Coates' and Perrin-Riou's paper On $p$-adic L-functions Attached to Motives over $\mathbb{Q}$ (see http://doi.org/10.2969/aspm/01710023), the ...
Maty Mangoo's user avatar
1 vote
0 answers
131 views

Zeroes of certain $L$-functions on the critical line and GGP conjectures

Global Gan-Gross-Prasad conjecture (on various groups) says that nonvanishing of certain automorphic $L$-function $L(s, \pi)$ (of cuspidal representation $\pi$ of some reductive group $G$) at $s = 1/2$...
Seewoo Lee's user avatar
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Whittaker function oscillation on diagonal near 0

In Proposition 1 of Blomer - Applications of the Kuznetsov formula on GL(3), bounds are given for the Whittaker function $$W_{\nu_1, \nu_2}(y_1, y_2) = \mathcal W_{\nu_1, \nu_2}\begin{pmatrix} y_1y_2 &...
Mayank Pandey's user avatar
6 votes
1 answer
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The error term for the second moment of Fourier coefficients of cusp forms with the level explicitly determined

There is a basis question which puzzles me for a while. The question is the following: Let $p$ be a prime and $X\ge 2$. Let $f$ be a $GL_2$-newform of level $p$ and non-trivial nebentypus, with the $n$...
hofnumber's user avatar
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3 votes
1 answer
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Residue of a local $\gamma$-factor and its relation with adjoint $\gamma$-factor

I met the following relation (if my understanding is correct) of local $\gamma$-factors when I was reading Hiraga-Ichino-Ikeda's paper "Formal Degrees and Adjoint $\gamma$-Factors": Let $F$ ...
youknowwho's user avatar
2 votes
0 answers
166 views

A list of $R=T$ theorems for $\mathbf{GSp}_4$

I know of only two cases of theorem $R=T$ where $T$ is a Hecke algebra acting on an automorphic forms (or representations) space of $\mathbf{GSp}_4$, the first one was proved by A.Genestier and J....
Marsault Chabat's user avatar
6 votes
1 answer
170 views

What is the conductor of an automorphic representation for $\Gamma_0(q)$ in $GSp(4)$?

Let $\pi$ be a generic cuspidal automorphic representation on $GSp(4)$, with level $\Gamma_0(q)$ (the group of symplectic matrices with lower left block divisible by $q$), i.e. $$\Gamma_0(q) = \left\{ ...
Desiderius Severus's user avatar
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Adelic functions of moderate growth

Let $f:GL_2(\mathbb A_\mathbb Q)\to\mathbb C$. For $g\in GL_2(\mathbb A)$ and a place $v$ of $\mathbb Q$, define $\|g\|_v=\max_{1\le i,j\le 2}(|g_{ij}|_v,|(g^{-1})_{ij}|_v).$ I have seen several ...
user14411's user avatar
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3 votes
1 answer
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Petersson norms of quaternionic modular forms

How is the Petersson norm of a quaternionic modular form defined? Background: In Tamiozzo, On the Bloch-Kato conjecture for Hilbert modular forms, section 3.3, it is written "We normalize $f_B$ ...
user471019's user avatar
2 votes
1 answer
145 views

What is the sum operation on torsors induced by Weil uniformization?

Let $k$ be an algebraically closed field, $G$ a reductive group, and $C$ a curve. The algebraic version of the Weil uniformization theorem (see e.g. arXiv:1511.06271v2) says that groupoid of $G$-...
Doron Grossman-Naples's user avatar
3 votes
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122 views

Modular forms on central division algebra of degree $\ge 3$

I just learned from some online sources including Buzzard's note and Emerton's answer on this MO question about quaternionic modular forms and explicit version of Jacquet-Langlands correspondence in ...
Seewoo Lee's user avatar
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1 vote
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what is the relationship betwen $L(s,sym^mf\times sym^mg)$ symmetric L function of $f$ and $g$ and $\lambda_{f}(n^m)$, $\lambda_{g}(n^m)$?

what is the relationship betwen $L(s,sym^mf\times sym^mg)$ symmetric L function of $f$ and $g$ and $\lambda_{f}(n^m)$, $\lambda_{g}(n^m)$ ?
Li Xnu's user avatar
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6 votes
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Divergence of integrals in the trace formula

I am trying to understand the following situation for $G=GL(2)$, when going from the compact trace formula to the non-compact case. The integral over $G(\mathbb{A})^1_\gamma \backslash G(\mathbb{A})^1$...
TheStudent's user avatar
2 votes
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Weil's "Sur la formule de Siegel dans la théorie des groupes classiques"

I am struggling to understand the following passage on Weil's text "Sur la formule de Siegel dans la théorie des groupes classiques" on page 16. If I understood correctly, in the second ...
Breakfastisready's user avatar
11 votes
1 answer
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Do people prefer working on $\mathrm{GSp}$ and $\mathrm{GU}$ rather than $\mathrm{Sp}$ and $\mathrm{U}$, and why?

I am a new learner of Iwasawa theory and currently reading the famous paper by Skinner-Urban in 2014, and the following-up works by many other people. When reading these papers, I found that some ...
Hetong Xu's user avatar
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8 votes
1 answer
240 views

Spectral decomposition of $\Gamma\backslash X$

Let $X$ be a reasonable manifold of non-positive curvature (could be $\mathbb{H}^n$, symmetric or locally symmetric space, homogeneous Hadamard manifold etc.), and let $\Gamma$ be a reasonable group ...
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