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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 ...
Krishnarjun's user avatar
2 votes
0 answers
84 views

Ordinary primes for a weak form corresponding to a CM newform

Setup: Let $f$ be a harmonic Maass form of weight $2-k$ ($k \in \mathbb{N}$), level $N$, and character $\chi$. Letting $q := e^{2\pi i z}$ and considering the Fourier expansion of any harmonic Maass ...
Freddie's user avatar
  • 26
8 votes
0 answers
151 views

Relation between the Laplace eigenvalue of Maass forms and the Virasoro algebra

I attended today a talk of Dorian Goldfeld. In the talk, he mentioned that for a Maass cusp form $\phi$ of $\mathrm{SL}(n, \mathbb{Z})$, with Langlands parameter $\alpha = \left( \alpha_1, \dots, \...
darkl's user avatar
  • 680
9 votes
1 answer
693 views

Spectral decomposition of product of modular functions

The eigenfunctions of the Laplacian on $SL(2,\mathbb Z)\backslash \mathbb H$ are known to be given by three types: the constant function, the real analytic Eisenstein series (which come in a ...
nathan benjamin's user avatar
7 votes
1 answer
261 views

Intuition about how Voronoi formulas change lengths of sums

In reading the literature one encounters countless examples of Voronoi formulas, i.e., formulas that take a sum over Fourier coefficients, twisted by some character, and controlled by some suitable ...
Jakob Streipel's user avatar
3 votes
0 answers
104 views

Maass forms associated with Ramanujan's mock theta functions

If you perform a fulltext search on the L-functions and modular forms database for "Ramanujan", you get entries related to the $\tau$ function and the modular discriminant $\Delta$. If you want the ...
graveolensa's user avatar
6 votes
1 answer
479 views

Analogues of Hecke relations for Maass forms

If a (suitably normalised) holomorphic cusp newform has q-expansion $$f(z) = \sum_n \lambda_f(n) e(nz),$$ then we know the Hecke relations for $(mn,q)=1$, $$(\star) \qquad \lambda_f(m)\lambda_f(n) = \...
TheStudent's user avatar
4 votes
1 answer
363 views

The exceptional eigenvalues and Weyl's law in level aspect

The Weyl law for Maass cusp forms for $SL_2(\mathbb Z)$ was obtained by Selberg firstly and has been generalized to various caces (Duistermaat-Guillemin, Lapid-Müller and others). For example, one of ...
Qinghua Pi's user avatar
3 votes
0 answers
197 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 $$\...
Desiderius Severus's user avatar
4 votes
2 answers
435 views

Nonvanishing of central L-values of Maass forms

Are there any results on the proportion of nonzero central L-values of Maass cusp forms? More precisely, I am looking for lower bounds for \begin{equation} \frac{\#\{\phi_j : \, L(1/2, \phi_j) \neq 0,...
João Guerreiro's user avatar
-1 votes
1 answer
507 views

Is Selberg's eigenvalue conjecture related to RH?

I took a quick glance on a survey paper about superzeta functions where one considers a pair $\rho\leftrightarrow 1-\rho$ of non trivial zeroes of the Riemann zeta function. The assumption of RH, i.e $...
Sylvain JULIEN's user avatar
2 votes
1 answer
446 views

Level dependence in the Ramanujan-Petersson Conjecture for GL(2) Maass forms

Suppose $f(z) = \sum_{n \geq 1} A(n)n^{\frac{k-1}{2}} e(nz)$ is a weight $k$ holomorphic cusp form on $\text{GL}(2)$. Then the Ramanujan-Petersson conjecture (proved in this case by Deligne) says ...
davidlowryduda's user avatar
2 votes
1 answer
433 views

Want more details about the image of a Maass form in the AIM press release concerning LMFDB

Actually I came upon this through MO a couple of days ago: in here (http://aimath.org/aimnews/lmfdb/) there is a mesmerizing image The caption reads A Maass form, one of the 20 different types of ...
მამუკა ჯიბლაძე's user avatar
2 votes
1 answer
516 views

Maass form properties and their fourier coefficients

Some Maass form can be written ($K_{iR}$ is the K-Bessel function): $$f(x+iy)=\sum_{n \ne 0}^{\infty} a_n \sqrt{y} \;K_{iR}(2\pi |n| y) \; e^{2 i\pi nx}$$ with the $a_n$ multiplicative, but inversly ...
Bertrand's user avatar
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6 votes
1 answer
596 views

Converse to Modularity II: Maass cusp forms

(This comes from this other question. You can find more details there) The following bijection is now a theorem: Odd irreducible 2-dim Galois repn $\longleftrightarrow$ weight 1 newforms note: ...
Myshkin's user avatar
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7 votes
1 answer
468 views

Complete L-function and FE of Rankin-Selberg on GL(2)?

Let $f$ be a Maass cusp form of $\Gamma_0(N)$ on the upper half plane with character $\chi$ mod $N$ and eigenvalue $1/4+\mu^2$. What is the complete $L$-function of the Rankin-Selberg product $L(s,f\...
7-adic's user avatar
  • 3,764
2 votes
1 answer
220 views

Infinite sum of asymptotic expansions

I have a question about an infinite sum of asymptotic expansions: Assume that $f_k(x)\sim a_{0k}+\dfrac{a_{1k}}{x}+\dfrac{a_{2k}}{x^2}+\cdots$ with $a_{0k}\leq \dfrac{1}{k^2}$, $a_{1k}\leq \dfrac{1}{k^...
Analysis's user avatar
2 votes
0 answers
234 views

Distribution of Fourier coefficients of Maass forms

In the sense of Maass an automorphic function $\phi$ with Laplace-Beltrami eigenvalue $\frac{(d-1)^2}{4}+t^2$ on $d$-dimensional hyperbolic space which can be thought as $\mathbb{R}^{d-1}\times\mathbb{...
Subhajit Jana's user avatar
7 votes
1 answer
295 views

Mean value of Maass forms

Let $X = SL_2(\mathbb{Z}) \backslash \mathbb{H}$ be the modular surface. Consider a basis of $L^2$-normalized Hecke-Maass cusps forms $\phi_j$ on $X$ with $-\Delta$-eigenvalue $\lambda_j$. Hejhal-...
Pig's user avatar
  • 809
5 votes
1 answer
574 views

Asymptotic behaviour of $K$-Bessel function in transition range

It is known that the famous mistake of Iwaniec-Sarnak in their paper of $L^\infty$ norm of eigenfunction of non-cocompact arithmetic surfaces in lemma (A1) is because of they did not consider the bump ...
Subhajit Jana's user avatar
7 votes
2 answers
1k views

Characterizing the real analytic Eisenstein series

Consider the classical real analytic Eisenstein series $$ E(z,s)=\left(\pi^{-s}\Gamma(s)\frac{1}{2}\right)\sum_{(m,n)\neq(0,0)}\frac{y^s}{|mz+n|^{2s}}, $$ where $z=x+iy$. We think of $E(z,s)$ as a ...
Hugo Chapdelaine's user avatar
6 votes
2 answers
560 views

Lower bound of Hecke eigenvalues of Maass form

If $f$ is a Maass form and $p$-Hecke eigenvalue (i.e. Hecke eigenvalue of usual Hecke operator $T_p$) of $f$ is $\lambda_f(p)$, do we know anything about lower bound of the sum$$S(x) = \sum_{x\le p\le ...
Subhajit Jana's user avatar
3 votes
0 answers
169 views

Asymptotic expansion of an integral, related to Maass forms

I am trying to compute the asymptotic expansion of the integral $I(t) = \int_{C} e^{\sqrt{1+u}(\frac{1}{t}+\frac{t^2}{\sqrt{u}})}\frac{u^\eta}{\sqrt{1+u}}du$ as $t$ is real and $t\rightarrow +\infty$, ...
Maass forms's user avatar
3 votes
1 answer
329 views

A database on Maass forms?

Is there somewhere a database on Maass forms that includes eigenvalues, Taylor coefficients, etc...? I am mainly interested in classical forms on $\Gamma(1)\backslash H$.
Alex Gavrilov's user avatar
11 votes
1 answer
2k views

Best record toward Selberg's eigenvalue conjecture?

What's the best record toward Selberg's eigenvalue conjecture: a Maass form on $\Gamma_0(N)$ has eigenvalue greater than or equal to 1/4?
7-adic's user avatar
  • 3,764
4 votes
2 answers
1k views

Mock Theta Functions

I am studying about Mock modular forms and Mock theta functions. I wonder how Zwegers connected mock theta functions with Harmonic Maass Forms? I mean, what was the philosophy/idea of Mock Theta ...
Subhajit Jana's user avatar
5 votes
2 answers
431 views

No Exceptional Eigenvalues of Weight 1/2 Maass Forms on $\Gamma_0(4)$?

Some colleagues and I were wondering if there is a citation out there which shows there are no exceptional eigenvalues, $\lambda$, of classical weight 1/2 Maass forms on $\Gamma_0(4)$, which is to say ...
Thomas Hulse's user avatar
3 votes
1 answer
409 views

Weyl law for SL(2,C)

Are there any estimates for the eigenvalues of the Laplace operator for $\Gamma \backslash SL(2, \mathbb{C})/SU(2)$ known beyond the main term? Here, $\Gamma$ should be congruence subgroup in $SL(2,o)$...
Marc Palm's user avatar
  • 11.1k
7 votes
4 answers
1k views

Multiplicity one conjecture

I recently became interested in Maass cusp forms and heared people mentioning a "multiplicity one conjecture". As far as I understood it, it says that the dimension of the space of Maass cusp form for ...
Ruedi Meier's user avatar
7 votes
1 answer
885 views

Are coefficients of Maass forms of eigenvalue 1/4 known to be algebraic?

I would really like to know whether the following famous conjecture has been solved. I've read in a few places that it has been solved, but I have been unable to find a reference. I do know that ...
John Pardon's user avatar
  • 18.3k
5 votes
3 answers
625 views

Equidistibution of horocycles through Hecke eigenvalues of Maass cusp forms

At the end of this very nice post: http://blogs.ethz.ch/kowalski/2012/05/21/who-needled-buffon/ E. Kowalski talks about the equidistribution of the points $\frac{j+i}{N}$ when $j=1,\dots,N$ and $N$ ...
Menny's user avatar
  • 638
28 votes
1 answer
3k views

Are there Maass forms where the expected Galois representation is $\ell$-adic?

Recall that by theorems of Deligne and Deligne--Serre, there is the following dichotomy: Modular forms on the upper half plane of level $N$ and weight $k\geq 2$ correspond to representations $\rho:\...
John Pardon's user avatar
  • 18.3k
14 votes
1 answer
1k views

Zeroes of Maass forms

By a Maass form I just mean--maybe a bit loosely--any real analytic $\Bbb C$-valued function $f$ on the upper halfplane $\cal{H}$ which is automorphic of weight $k\in\Bbb Z$ with respect to a discrete ...
Andrea Mori's user avatar