# Questions tagged [maass-forms]

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33
questions

2
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0
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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 ...

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 ...

8
votes

0
answers

151
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### 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, \...

9
votes

1
answer

693
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### 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 ...

7
votes

1
answer

261
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### 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 ...

3
votes

0
answers

104
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### 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 ...

6
votes

1
answer

479
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### 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) = \...

4
votes

1
answer

363
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### 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 ...

3
votes

0
answers

197
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### 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
$$\...

4
votes

2
answers

435
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### 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,...

-1
votes

1
answer

507
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### 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 $...

2
votes

1
answer

446
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### 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 ...

2
votes

1
answer

433
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### 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 ...

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 ...

6
votes

1
answer

596
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### 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: ...

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\...

2
votes

1
answer

220
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### 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^...

2
votes

0
answers

234
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### 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{...

7
votes

1
answer

295
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### 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-...

5
votes

1
answer

574
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### 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 ...

7
votes

2
answers

1k
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### 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 ...

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 ...

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 $t\rightarrow +\infty$, ...

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 $\Gamma(1)\backslash H$.

11
votes

1
answer

2k
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### 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?

4
votes

2
answers

1k
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### 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 ...

5
votes

2
answers

431
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### 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 ...

3
votes

1
answer

409
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### 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 $SL(2,o)$...

7
votes

4
answers

1k
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### 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 ...

7
votes

1
answer

885
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### 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 ...

5
votes

3
answers

625
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### 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 $N$ ...

28
votes

1
answer

3k
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### 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:\...

14
votes

1
answer

1k
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### 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 ...