Skip to main content

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.

Filter by
Sorted by
Tagged with
4 votes
1 answer
145 views

Is "self-dual" equivalent to "dihedral" for Maass forms on $\mathrm{GL}(2)$?

Suppose $f$ is a Maass cusp form on $\Gamma_0(D)$. The associated symmetric square $L$-function $L(\mathrm{sym}^2 f, s)$ has a pole at $s = 1$ if and only if $f = \overline{f}$ (if $f$ is self-dual). ...
davidlowryduda's user avatar
8 votes
1 answer
521 views

Roadmap to Carayol-Deligne-Langlands

Having begun self-study of Fermat's Last Theorem a few years ago, I have only recently begun to understand and appreciate the theorem of Carayol-Deligne-Langlands on local-global compatibility for ...
Johnny Apple's user avatar
1 vote
0 answers
104 views

Notion of "Hodge bundle" for abelian type Shimura varieties

For a Siegel type Shimura datum $(\text{GSp}_{2g}, \mathcal{H}^{\pm})$ and level $K$, we construct the Shimura variety $S_{g,K} := \text{Sh}_K(\text{GSp}_{2g},\mathcal{H}^{\pm})$. We have a universal ...
ChimiSeanGa's user avatar
3 votes
1 answer
175 views

Counting local representations for $\mathrm{GL}_2$

$\DeclareMathOperator\GL{GL}$Some context. In number theory, it is natural to study distribution questions for the family of elliptic curves over $\mathbb{Q}$ (or any fixed number field for that ...
Anwesh Ray's user avatar
3 votes
1 answer
148 views

The lower bound for the automorphic $L$-function $L(s,\pi)$ at the edge of the critical strip $\Re s=1$

Let $\pi$ be any automorphic Maass form on $\text{GL}_m$ of level $N$, say. Assume that the associated $L$-function $L(s,\pi)$ satisfies some good conditions; for example, it satisfies the functional ...
user528074's user avatar
5 votes
1 answer
126 views

A question on hybrid subconvexity for individual L-functions

Sorry to disturb. I have a question need some explanations from the experts on the MO-website. As usual, we let $L(f,s)$ be the corresponding $L$-function associated to the newform $f$ on $SL_2(\...
user528074's user avatar
3 votes
0 answers
63 views

shifted convolution in arithmetic progressions

Let $r(n)$ be the number of ways of writing $n$ as the sum of two integer squares. Asymptotics for the shifted convolution problem $$ \sum_{n\in \mathbb N\cap[1,x]}r(n) r(n+1)$$ are quite classical; a ...
Dr. Pi's user avatar
  • 2,999
2 votes
0 answers
70 views

Number of rational points of a connected reductive group in a compact subset

Let $G$ be a connected reductive $\mathbb{Q}$-group. Let $\mathbb{A}$ denote the ring of adèles of $\mathbb{Q}$. Let $B \subset G(\mathbb{A})$ be a compact, let $x \in G(\mathbb{A})$ and consider the ...
Sentem's user avatar
  • 31
5 votes
0 answers
116 views

Rankin-Selberg convolutions with mixed integral and half-integral weights

Let $f(z)$ denote a weight $0$ Hecke-Maass form of level $N$ and let $\theta(z)$ denote the Jacobi theta function. Then $y^{1/4} f(z) \overline{\theta(z)}$ transforms as an automorphic form of weight $...
awwalker's user avatar
4 votes
1 answer
126 views

Reference Request: Test vectors for local Rankin-Selberg L-factors in ramified cases

Let $F$ be a global number field, i.e. a finite extension of the field of rational numbers. Let $\sigma$, $\pi$ be automorphic representations of $\mathrm{GL}_n(F)$ and $\mathrm{GL}_{n+1}(F)$ ...
Hetong Xu's user avatar
  • 619
3 votes
0 answers
71 views

Logarithm map for groups defined over adelic ring

I've been reading the book Eisenstein series and automorphic representations and I am struggling to understand the definition of a logarithm map $H:G(\mathbb{A})\rightarrow \mathfrak{h}(\mathbb{R})$ (...
Ji Woong Park's user avatar
1 vote
0 answers
44 views

Genericity of local representation with a non-generic local A-parameter

Let $\pi$ be an irreducible smooth representation of a classical $p$-adic group. Suppose that $\pi$ has a local L-parameter associated to some non-generic local A-parameter $\psi$. Then I am wondering ...
Andrew's user avatar
  • 939
3 votes
0 answers
86 views

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 $T_n$ ...
Andrew's user avatar
  • 939
1 vote
0 answers
125 views

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
  • 323
2 votes
1 answer
87 views

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 $n \geq m$. ...
Hetong Xu's user avatar
  • 619
2 votes
0 answers
54 views

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 ...
Andrew's user avatar
  • 939
2 votes
0 answers
74 views

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=...
Andrew's user avatar
  • 939
1 vote
0 answers
98 views

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|\...
Andrew's user avatar
  • 939
6 votes
1 answer
442 views

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 ...
user15243's user avatar
  • 474
1 vote
1 answer
137 views

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
  • 619
6 votes
1 answer
491 views

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 ...
stupid_question_bot's user avatar
3 votes
0 answers
111 views

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
3 votes
0 answers
104 views

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
  • 619
2 votes
0 answers
136 views

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 ...
LWW's user avatar
  • 663
3 votes
1 answer
200 views

$p$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
  • 663
2 votes
1 answer
242 views

'$\times$' 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
3 votes
0 answers
103 views

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
  • 553
4 votes
1 answer
440 views

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
  • 474
2 votes
1 answer
130 views

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
  • 553
5 votes
1 answer
273 views

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
1 vote
1 answer
177 views

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
  • 939
1 vote
0 answers
112 views

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
  • 939
0 votes
1 answer
208 views

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 ...
user avatar
1 vote
0 answers
165 views

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
  • 553
1 vote
0 answers
127 views

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
  • 939
6 votes
0 answers
373 views

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
2 votes
1 answer
222 views

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
4 votes
1 answer
261 views

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
  • 553
4 votes
1 answer
269 views

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
5 votes
1 answer
482 views

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
2 votes
0 answers
195 views

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
  • 91
4 votes
0 answers
180 views

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
3 votes
1 answer
221 views

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
  • 525
7 votes
1 answer
256 views

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
3 votes
0 answers
86 views

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
  • 1,701
2 votes
1 answer
189 views

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
  • 939
6 votes
0 answers
227 views

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
132 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
  • 1,961
2 votes
0 answers
104 views

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
289 views

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
  • 553

1
2 3 4 5
16