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).

two types of Maass forms

• square integrable Maass forms ~ discrete spectrum
• Eisenstein series ~ continuous spectrum

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

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

examples

Eisenstein series

• 틀:수학노트

Maass-Poincare series

• Hejhal
• real analytic eigenfunction of the Laplacian with known singularities at \(i\infty\)

 

Kloosterman sum

• 틀:수학노트

 

related items

• harmonic Maass forms
• spectral theory of automorphic forms
• q-series and Maass forms 

 

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)

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxMll2Y2lzeWg0UVE/edit
• http://www.math.chalmers.se/~sj/forskning/level11_2a.pdf
• http://www.ijpam.eu/contents/2009-54-2/12/12.pdf

encyclopedia

• http://en.wikipedia.org/wiki/Kronecker_limit_formula
• http://en.wikipedia.org/wiki/Real_analytic_Eisenstein_series

question and answers(Math Overflow)

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

expositions

• Jianya Liu LECTURES ON MAASS FORMS

articles

• Strömberg, Fredrik. “Computation of Maass Waveforms with Nontrivial Multiplier Systems.” Mathematics of Computation 77, no. 264 (2008): 2375–2416. doi:10.1090/S0025-5718-08-02129-7.