Maass forms

introduction

Hyperbolic distribution problems and half-integral weight Maass forms
Automorphic forms correspond to representations that occur in $L_2(\Gamma\backslash G)$.
In the case when $G$ is $SL(2,\mathbb{R})$
- holomorphic modular forms correspond to (highest weight vectors of) discrete series representations of $G$
- Maass wave forms correspond to (spherical vectors of) continuous series representations of G.

definition

A Maass (wave) form = continuous complex-valued function f of τ = x + iy in the upper half plane satisfying the following conditions:
- f is invariant under the action of the group SL₂(Z) on the upper half plane.
- f is an eigenvector of the Laplacian operator $\Delta=-y^2\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right)$
- f is of at most polynomial growth at cusps of SL₂(Z).
Maass-Poincare series
- Hejhal
- real analytic eigenfunction of the Laplacian with known singularities at $i\infty$

fourier expansion

$f(z+1)=f(z)$ and $\Delta f=\lambda f$ where $\lambda = s(1-s)$ and $\Re s \geq 1/2$ imply

$$ f(x+iy)=\sum_{n\in \mathbb{Z}}a_n \sqrt{y}K_{s-1/2}(2\pi |n| y) e^{2\pi i n x} $$ where $K_{\nu}$ is the modified Bessel function of the second kind

under the assumption that $f(x+iy)=f(-x+iy)$, we get

$$ f(x+iy)=\sum_{n=1}^{\infty}a_n \sqrt{y}K_{s-1/2}(2\pi n y) \cos (2\pi i n x) $$

Eisenstein series

틀:수학노트

Kloosterman sum

틀:수학노트

related items

books

Henryk Iwaniek, Emmanuel Kowalski (2004). Analytic number theory
Lectures on modular functions of one complex variable (Tata Institute of Fundamental Research. Lectures on mathematics and physics. Mathematics, 29)
- Hans Maass, (pdf)

expositions

Jianya Liu LECTURES ON MAASS FORMS

computational resource

encyclopedia

question and answers(Math Overflow)

http://mathoverflow.net/questions/52744/what-is-the-relationship-between-modular-forms-and-maass-forms