All Questions
Tagged with automorphic-forms modular-forms
60 questions with no upvoted or accepted answers
12
votes
0
answers
508
views
Weyl law for Maass forms with nontrivial character
The classical Weyl law for $\Gamma = \mathrm{SL}_2(\mathbb{Z})$ counts the number of Maass cusp forms on $\Gamma \backslash \mathbb{H}$ with Laplace eigenvalue less than $T$. This is originally due to ...
- 8,374
11
votes
0
answers
239
views
Representation of the space of lattices in $\Bbb R^n$
The space of 2D lattices in $\Bbb R^2$ can be represented with the two Eisenstein series $G_4$ and $G_6$. Each lattice uniquely maps to a point in $\Bbb C^2$ using these two invariants, and the points ...
- 4,897
11
votes
0
answers
231
views
Eisenstein series for non congruence subgoups
What is the present status of the Eisenstein series for noncongruence subgroups?
I am aware of work of A. Scholl and Rohrlich work on the subject.
Is there any specific examples that has been ...
- 248
9
votes
0
answers
147
views
Terminology question for the cohomology of the Hilbert modular group
Let $\Gamma$ be the Hilbert modular group of determinant one matrices with entries in the ring of integers of a real quadratic field $F$, and let $M$ be a $\Gamma$-module. Is there a standard name for ...
- 91
9
votes
0
answers
211
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},\...
- 562
9
votes
0
answers
435
views
Rankin-Selberg for Maass form GL(3)xGL(2)
Let $F$ be a Maass cusp form for $\mathrm{SL}(3,\mathbb{Z})$ (level 1 trivial character).
Let $g$ be a Maass cusp form for $\Gamma_0(N)$ with character $\chi$ mod $N$. For convenience, you may assume ...
- 3,804
8
votes
1
answer
356
views
Average bounds on Rankin-Selberg coefficients for modular forms
Let $f$ be a cuspidal Hecke newform of weight $k$ and level $N$, and denote by $a_f(n)$ its $n$-th Fourier coefficient. The newform $f$ is normalized so that $a_f(1) = 1$. As a consequence of Rankin-...
- 5,424
7
votes
0
answers
122
views
Theta Function Associated to Kummer Lattice
This is something which I feel must be out in the literature somewhere, but I have been unable to find anything.
If we let $\text{Km}(A)$ be the Kummer $K3$ surface associated to an abelian surface $A$...
- 1,701
6
votes
0
answers
236
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
$$f(gz) = f(z) (cz + d)^{2k} \qquad \text{for} \...
- 4,164
6
votes
0
answers
1k
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What is the best way to learn about Modular Forms?
I am a senior Mathematics Major, and I am interesting in learning about Modular Forms. I have a layman's general sense of what they are but I was wondering if there is a lecture(I am willing to pay) ...
6
votes
0
answers
339
views
Is there an integral pairing between quaternionic Hecke algebras and cusp forms?
Let $F$ be a totally real number field with integers $\mathcal{O}_F$ and $B$ a quaternion algebra over $F$ split at exactly one infinity place.Fix $n\geq 1$ and like in the special case $F=\mathbb{Q}, ...
- 469
5
votes
0
answers
1k
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Some questions about cuspidal representations and automorphic representations
My reference is Daniel Bump's book, Automorphic Forms and Representations. $G$ is a connected reductive group over a number field $k$ (in Bump's book he takes $G = \operatorname{GL}_n$). Let $K = K_{...
- 6,180
5
votes
0
answers
201
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The Geometry of Jacobi Forms and their Asymptotic Expansions
A Jacobi form of weight $k$ and index $m$ is a meromorphic function $\varphi: \mathbb{H} \times \mathbb{C} \to \mathbb{C}$ satisfying
$$\varphi\bigg(\frac{a \tau + b}{c \tau + d}, \frac{z}{c \tau + d}...
- 1,701
5
votes
0
answers
128
views
Volumes of Hecke operators
Let $G=GL(2, F)$ and $K$ a maximal compact subgroup. Unramified Hecke operators are defined by the action of the double cosets
$$T(n) = \bigcup_{\substack{ad=n, a>0 \\ a|d}} K \left( \begin{array}{...
- 551
5
votes
0
answers
133
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Modularity of the Shimura-Shintani kernel
Following this paper by Bringmann, Kane, Kohnen, the kernel of the Shimura and Shintani lifts is the function $$\Omega(\tau,z) = \frac{1}{\binom{2k-2}{k-1}\pi} \sum_{D=1}^{\infty} \sum_{b^2 - 4ac = D} ...
- 51
5
votes
0
answers
139
views
Status of representation theory of harmonic weak Maass forms
It is well-known that the classical Hecke Maass/modular forms can be reinterpreted as vectors in irreducible automorphic representations of GL(2). On the contrary, for weak Maass form/weakly ...
- 809
5
votes
0
answers
439
views
different definitions of epsilon constants for representations of GL(2) from modular forms
I ran into this question when trying to compute the Atkin-Lehner pseudo-eigenvalue of
newforms. Let $k \geq 2$, let $\omega$ be a Dirichlet character modulo $N$ and let $f \in S_k(N,\omega)$ be a ...
- 221
5
votes
0
answers
292
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Voronoi formula and twists by additive characters
I was wondering if there are any references for the error term in the problem
$$\sum_{n\leq x} r(n) \exp(2\pi i\frac{a}{q}n)$$
where $r(n)$ is the number of representations of $n$ as a sum of two ...
- 3,062
4
votes
0
answers
103
views
Automorphic forms on $\mathrm{GL}_{2}$, $\mathrm{SL}_{2}$, and $\mathrm{Mp}_{2}$ — classical counterparts
I asked exact same question on MSE but haven't got answer yet, so asking here, too. I may erase the original one once I got an answer here.
--
I'm confusing about automorphic representations of $\...
- 2,215
4
votes
0
answers
187
views
Understanding Shimura correspondence in context of Langlands functoriality
Recently, I started to read about automorphic forms and representations on covering groups, e.g. metaplectic groups. I set my first goal as understanding Shimura's correspondence in representation ...
- 2,215
4
votes
0
answers
155
views
How to obtain the harmonic theta series via the global theta correspondence explicitly?
I am interested in the following kind of modular forms: let $(L,q)$ be an even unimodular lattice inside $V:=L\otimes \mathbb{Q}$, and $P$ is a harmonic homogeneous polynomial of degree $d$ on $\...
- 391
4
votes
0
answers
204
views
A question on the twisted symmetric square L-functions
Sorry to disturb. I have a puzzle which might be naive for many experts here.
Let $f$ be a Hecke newform of prime level $N$ on $\mathrm{GL}_2$, and $
\chi$ a primitive character of square-free ...
- 353
4
votes
0
answers
135
views
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 ...
- 563
4
votes
0
answers
167
views
How to express the cuspidal form in terms of Poincare series?
Sorry to disturb. I recently read Blomer's paper. There is a blur which needs the expert's help here. Blomer's paper says "For any cusp form $f$ of integral weight $k$, level $N$, $f$ can be written ...
- 353
4
votes
0
answers
169
views
Hecke operators that lower level
I am working with weakly holomorphic modular functions (weight $=0$) $f \in M_0(\Gamma(N), \chi)$ of level $N$ with some character $\chi$ (We can ignore the character for now). Let $f \in M_0(\Gamma(N)...
user35360
3
votes
0
answers
190
views
Voronoi formula on $\mathrm{GL}_4$ in the level aspect with ramification
$\DeclareMathOperator\GL{GL}$Let $f$ be an automorphic form on $\GL_3$ for $\Gamma_0(p)$ with $p$ being a prime (see Bump or Goldfeld's books for definitions). Recall that, in this paper-"The ...
- 41
3
votes
0
answers
122
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 (...
3
votes
0
answers
96
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 $\Gamma _{0}^{(2)}(N) \subset \text{Sp} _{4}(...
- 1,701
3
votes
0
answers
144
views
Ash–Stevens for Hilbert modular forms
In the theory of mod-$p$ modular forms, I learned a while ago about an interesting result that I think is technically due to Serre and Tate, though the proof was first published by Jochnowitz in ...
- 241
3
votes
0
answers
89
views
Hoffstein–Lockhart for non-congruence subgroups
Let $\Gamma$ be a non-congruence subgroup of $\operatorname{SL}(2,\mathbb{Z})$ of finite index and let $f$ be a holomorphic cuspidal modular form of weight $k$ for the group $\Gamma$. For simplicity, ...
- 589
3
votes
0
answers
95
views
Principal series representation of $SL(2,\mathbb{R})$ restricted to principal congruence subgroup
Given a principal congruence subgroup $\Gamma(N)$ of $SL(2,\mathbb{R})$, since $\Gamma(N)$ is free, consider a probability distribution $\mu_1$ of a simple random walk on $\Gamma(N)$ and consider its ...
- 131
3
votes
0
answers
77
views
Algebraic independece of modular forms for Fricke group over $\mathbb C(q)$
Let $q=e^{\pi i \tau}$ for $\tau \in \mathbb H$ and
$$
P_2(q)=\frac{P(q)+2P(q^2)}{3}, \quad Q_2(q)=\frac{Q(q)+4Q(q^2)}{5}, \\ R_2(q)=\frac{R(q)+8R(q^2)}{9},
$$
a Fourier series of quasi-modular form ...
- 663
3
votes
0
answers
128
views
Is constant map from automorphic form surjective?
Let $G$ be a connected reductive group over $\mathbb{Q}$ and $P=NM$ be a standard parabolic subgroup of $G$ and and $K$ a 'good' maximal compact subgroup of $G$. (For precise definitions of these ...
- 1,759
3
votes
0
answers
166
views
Automorphy Factor from Vector Bundles on Compact Dual
So I'm coming from an algebraic geometry perspective and I'm trying to carefully piece together the story of interpreting automorphic forms as sections of vector bundles on Shimura varieties. I think ...
- 1,701
3
votes
0
answers
156
views
Probability distribution from equidistribution - I
Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$. Denote $N_r(a,b)$ to be minimum $\ell_r$ norm of vector $(x,y)$ as $(x,y)$ ranges over all non-zero integral solutions to $(x,y)\equiv t(a,...
- 13.9k
3
votes
0
answers
299
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 ...
- 2,044
3
votes
0
answers
203
views
Expression of the root number for Maass forms
Take a holomorphic cusp newform, say $f \in S_k(N)^\mathrm{new}$, for a squarefree level $N$. It is an eigenvalue of the Atkin-Lehner operator, and this feature allows to express its root number as
$$\...
- 5,424
3
votes
0
answers
222
views
Functoriality for triple product GL(2) x GL(2) x GL(2)
Let $f$, $g$ and $h$ be three general automorphic forms on $\operatorname{GL}(2)$.
Do we know that $L(s, f\times g\times h)$ comes from an automorphic form on $\operatorname{GL}(8)$?
- 3,804
2
votes
0
answers
164
views
Bounds of modular functions on the Ford circles
Assume a holomorphic function from a product of two upper half planes $Z: \mathbb{H}_+\times \mathbb{H}_+\rightarrow \mathbb{C}$ with an expansion of the form
$$
Z(\tau,\tau') = \sum_{(h,h')\in S} a_{...
- 21
2
votes
0
answers
98
views
Extrema of real analytic Eisenstein series and more general modular functions
The real analytic Eisenstein series defined by the Poincare sum
$$E(s,z)=\sum_{(m,n)\neq (0,0)} {y^s\over |mz+n|^{2s}}$$
for $z\in{\mathbb H}$ and ${\rm Re}(s)>1$ is a manifestly $SL(2,{\mathbb Z})$...
- 21
2
votes
0
answers
192
views
Modular forms and Petersson inner product via De Rham cohomology, Hodge filtration and cup products
I'm looking for an explanation on how and why you can define modular forms through De Rham cohomology via the Hodge filtration and especially how the Petersson inner product is related to the cup ...
- 804
2
votes
0
answers
205
views
Two basic questions on congruence subgroups
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$I have two questions related to congruence subgroups.
Let $$\Gamma=\Gamma_0(N)=\Big\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \...
- 1,019
2
votes
0
answers
179
views
On Fourier coefficients of Bianchi modular forms, l-ordinary
Let $f\in S_2(\Gamma_1(N))$ be a Hecke eigenform and $\ell$ a prime number does not divide $N$. Let $a_f(\ell)$ be the $\ell$-th Fourier coefficient of $f$. Then $a_f(\ell)$ is is called $\ell$-...
- 21
2
votes
0
answers
107
views
Paramodular forms with level and Iwahori subgroups?
Given an integer $N>0$, not necessarily prime, we have the paramodular group $K(N) \subset \text{Sp}_{4}(\mathbb{Q})$, which consists of matrices of the form
$$\begin{bmatrix} * & *N & * &...
- 1,701
2
votes
0
answers
105
views
Possible Context for this "Siegel-like" Modular Form Construction?
The following construction of something very nearly a Siegel modular form of degree 2 arose in my research. I'm outside the worlds of automorphic forms and number theory, so I'm wondering if it ...
- 1,701
2
votes
0
answers
117
views
automorphic form associated with Apollonian Gasket
In /Indra's Pearls/, it's mentioned one can associate automorphic forms with limit sets. Is there an explicit description of the one associated with the Apollonian gasket (up to some appropriate ...
- 975
2
votes
0
answers
50
views
Product of $V_N$ operator(index changing) and its adjoint on Jacobi forms
In Kohnen & Skoruppa's 1989 inventiones paper, page 549, the operator $V_N: J_{k,1}^\text{cusp} \longrightarrow J_{k,N}^\text{cusp}$ is defined by the action
$$ \sum_{\substack{D<0,r \in \...
- 367
2
votes
0
answers
78
views
automorphic forms associated with symmetries of vertices of uniform honeycombs in hyperbolic space
Is there a catalogue of automorphic forms (modular/Maass/Siegel/Hilbert...) which lists them in terms of Poincaré series associated with the symmetries of the vertices of uniform honeycombs in ...
- 975
2
votes
0
answers
159
views
The dimension of the space of automorphic forms with multiplier system
Let $\Gamma$ be a discrete subgroup of $SL_{2}(\mathbb{Z})$ and $\vartheta$ a multiplier system of weight $k$ for $\Gamma$, by which we mean a function $\vartheta:\Gamma \rightarrow \mathbb{C}$ ...
- 433
2
votes
0
answers
212
views
Conceptual reason behind Shimura lifts
Shimura lifts are correspondence between integer weight and half-integral weight automorphic forms. Half integral weight things are associated to representation of a double cover of $G =SL_2(\mathbb{R}...
- 11.2k