Maass forms

==introduction

• Hyperbolic distribution problems and half-integral weight Maass forms
• Automorphic forms correspond to representations that occur in L2(G/Γ). In the case when G is SL2, holomorphic modular forms correspond to (highest weight vectors of) discrete series representations of G, while Maass wave forms correspond to (spherical vectors of) continuous series representations of G.

 

==Eisenstein series

• z = x + iy in the upper half-plane
• Re(s) > 1
• definition
  \(E(z,s) ={1\over 2}\sum_{(m,n)=1}{y^s\over|mz+n|^{2s}}\)
  
• Maass form
  \(\Delta E(z,s)=-y^2\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)E(z,s) = s(1-s)E(z,s)\)
  
• functional equation
  \(E^{*}(z,s) = \pi^{-s}\Gamma(s)\zeta(2s)E(z,s) \)
  \(E^{*}(z,s)=E^{*}(z,1-s)\)
  
• a unique pole of residue 3/π at s = 1

 

 

==Kloosterman sum

• used to estimate the Fourier coefficients of modular forms
• definition for prime p
  \(S(a,b;p)=\sum_{1\leq x\leq p-1}{\exp(2i\pi (ax+b\bar{x})/p)},\quad\text{where}\quad x\bar{x}\equiv 1\text{ mod } p\)
  
• generally defined as
  \(K(a,b;m)=\sum_{0\leq x\leq m-1,\ gcd(x,m)=1 } e^{2\pi i (ax+bx^*)/m}\)
  
• http://blogs.ethz.ch/kowalski/2010/02/26/the-fourth-moment-of-kloosterman-sums/
• Kloosterman, H. D. On the representation of numbers in the form ax² + by² + cz² + dt², Acta Mathematica 49 (1926), pp. 407-464

 

 

==history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

==related items

• examples of harmonic 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)
• 찾아볼 수학책
  
• http://gigapedia.info/1/Iwaniek
• http://gigapedia.info/1/Maass

 

 

==encyclopedia

• [1]http://en.wikipedia.org/wiki/Kronecker_limit_formula
• http://en.wikipedia.org/wiki/Real_analytic_Eisenstein_series
• http://en.wikipedia.org/wiki/Kloosterman_sum

 

 

==question and answers(Math Overflow)

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