Questions tagged [maass-forms]
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37 questions
3
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Why the hyperbolic Laplacian?
In the theory of automorphic forms there is the weight $k\in\mathbb{Z}$ Laplacian
\begin{align*}
\Delta_k:=-y^2 \left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)+iky\frac{\...
- 177
5
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0
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138
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Compute weight of modular form from its Fourier coefficients
It is known that Hecke eigenform $f \in S_{k}(\Gamma_0(N), \chi)$ is uniquely determined by first $C_{k,N}$ many Fourier coefficients, where $C_{k,N}$ is a constant only depends on $k$ and $N$. For ...
- 2,215
4
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1
answer
201
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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).
...
- 1,570
5
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0
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132
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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 $...
- 51
2
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0
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117
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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 ...
- 589
2
votes
0
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86
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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 ...
- 26
8
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157
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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, \...
- 730
9
votes
1
answer
750
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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
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1
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314
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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 ...
- 198
3
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107
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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 ...
- 975
6
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1
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557
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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 $(mn,q)=1$,
$$(\star) \qquad \lambda_f(m)\lambda_f(n) = \...
- 927
4
votes
1
answer
374
views
The exceptional eigenvalues and Weyl's law in level aspect
The Weyl law for Maass cusp forms for $SL_2(\mathbb Z)$ was obtained by Selberg firstly and has been generalized to various caces (Duistermaat-Guillemin, Lapid-Müller and others). For example, one of ...
- 143
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203
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Expression of the root number for Maass forms
Take a holomorphic cusp newform, say $f \in S_k(N)^\mathrm{new}$, for a squarefree level $N$. It is an eigenvalue of the Atkin-Lehner operator, and this feature allows to express its root number as
$$\...
- 5,424
4
votes
2
answers
447
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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
570
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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 $...
- 6,976
2
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1
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493
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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 ...
- 1,570
2
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1
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442
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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 ...
- 18.4k
2
votes
1
answer
566
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 ...
- 1,199
6
votes
1
answer
612
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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: ...
- 17.6k
7
votes
1
answer
488
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\...
- 3,804
2
votes
1
answer
223
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^...
- 21
2
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0
answers
239
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{...
- 1,692
7
votes
1
answer
300
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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-...
- 809
5
votes
1
answer
598
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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 ...
- 1,692
7
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2
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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 ...
- 7,609
6
votes
2
answers
595
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 ...
- 1,692
3
votes
0
answers
170
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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 $t\rightarrow +\infty$, ...
- 31
3
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1
answer
334
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 $\Gamma(1)\backslash H$.
- 6,891
12
votes
1
answer
3k
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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?
- 3,804
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 ...
- 1,692
5
votes
2
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465
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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 ...
- 53
3
votes
1
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430
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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 $SL(2,o)$...
- 11.2k
7
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4
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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 ...
- 123
7
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1
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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 ...
- 18.7k
5
votes
3
answers
632
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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 $N$ ...
- 638
28
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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:\...
- 18.7k
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 ...
- 797