# Questions tagged [maass-forms]

The tag has no usage guidance.

33 questions
Filter by
Sorted by
Tagged with
104 views

### 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 ...
• 537
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 ...
• 26
151 views

• 6,922
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 ...
• 1,478
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 ...
• 17.6k
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 ...
• 1,121
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: ...
• 17.5k
468 views

• 21
234 views

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 $\mathbb{R}^{d-1}\times\mathbb{... • 1,682 7 votes 1 answer 295 views ### Mean value of Maass forms Let$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$. Hejhal-... • 809 5 votes 1 answer 574 views ### Asymptotic behaviour of$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 ... • 1,682 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 ... • 7,531 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 ... • 1,682 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$\Gamma(1)\backslash H$. • 6,861 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? • 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 ... • 1,682 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 ... 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$SL(2,o)$... • 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 ... • 123 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 ... • 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$N$... • 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:\...
• 18.3k
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 ...