"Maass forms"의 두 판 사이의 차이
둘러보기로 가기
검색하러 가기
imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
||
| 2번째 줄: | 2번째 줄: | ||
* Hyperbolic distribution problems and half-integral weight Maass forms | * Hyperbolic distribution problems and half-integral weight Maass forms | ||
| − | * Automorphic forms correspond to representations that occur in | + | * Automorphic forms correspond to representations that occur in <math>L_2(\Gamma\backslash G)</math>. |
| − | * In the case when | + | * In the case when <math>G</math> is <math>SL(2,\mathbb{R})</math> |
| − | ** holomorphic modular forms correspond to (highest weight vectors of) discrete series representations of | + | ** holomorphic modular forms correspond to (highest weight vectors of) discrete series representations of <math>G</math> |
** Maass wave forms correspond to (spherical vectors of) continuous series representations of G. | ** Maass wave forms correspond to (spherical vectors of) continuous series representations of G. | ||
| 22번째 줄: | 22번째 줄: | ||
==fourier expansion== | ==fourier expansion== | ||
| − | * | + | * <math>f(z+1)=f(z)</math> and <math>\Delta f=\lambda f</math> where <math>\lambda = s(1-s)</math> and <math>\Re s \geq 1/2</math> imply |
| − | + | :<math> | |
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} | 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} | ||
| − | + | </math> | |
| − | where | + | where <math>K_{\nu}</math> is the modified Bessel function of the second kind |
| − | * under the assumption that | + | * under the assumption that <math>f(x+iy)=f(-x+iy)</math>, we get |
| − | + | :<math> | |
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) | 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) | ||
| − | + | </math> | |
| 60번째 줄: | 60번째 줄: | ||
* Henryk Iwaniek, Emmanuel Kowalski (2004). Analytic number theory | * 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) | + | * Lectures on modular functions of one complex variable (Tata Institute of Fundamental Research. Lectures on mathematics and physics. Mathematics, 29) |
** Hans Maass, ([[5323613/attachments/3133467|pdf]]) | ** Hans Maass, ([[5323613/attachments/3133467|pdf]]) | ||
2020년 11월 14일 (토) 01:25 판
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
Maass-Poincare series
- Hejhal
- real analytic eigenfunction of the Laplacian with known singularities at \(i\infty\)
Kloosterman sum
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)
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.