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.lmfdb.org/ModularForm/GL2/Q/Maass/
• http://www.math.chalmers.se/~sj/forskning/level11_2a.pdf
• http://www.ijpam.eu/contents/2009-54-2/12/12.pdf
• https://mathoverflow.net/questions/22908/does-anyone-want-a-pretty-maass-form

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

• Farrell Brumley, Simon Marshall, Lower bounds for Maass forms on semisimple groups, arXiv:1604.02019 [math.NT], April 07 2016, http://arxiv.org/abs/1604.02019
• 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.
• Booker, Andrew R., Andreas Strömbergsson, and Akshay Venkatesh. “Effective Computation of Maass Cusp Forms.” International Mathematics Research Notices 2006 (January 1, 2006): 71281. doi:10.1155/IMRN/2006/71281.

메타데이터

위키데이터

• ID : Q6721246

Spacy 패턴 목록

• [{'LOWER': 'maass'}, {'LOWER': 'wave'}, {'LEMMA': 'form'}]
• [{'LOWER': 'maass'}, {'LOWER': 'cusp'}, {'LEMMA': 'form'}]
• [{'LOWER': 'maass'}, {'LEMMA': 'form'}]