"Maass forms"의 두 판 사이의 차이
둘러보기로 가기
검색하러 가기
imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
||
| (사용자 2명의 중간 판 14개는 보이지 않습니다) | |||
| 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. | ||
| − | + | ||
| + | ==definition== | ||
| − | ==Eisenstein series== | + | * A Maass (wave) form = continuous complex-valued function <em style="">f</em> of τ = <em style="">x</em> + <em style="">iy</em> in the upper half plane satisfying the following conditions: |
| + | ** <em style="">f</em> is invariant under the action of the group SL<sub style="line-height: 1em;">2</sub>('''Z''') on the upper half plane. | ||
| + | ** <em style="">f</em> is an eigenvector of the Laplacian operator <math>\Delta=-y^2\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right)</math> | ||
| + | ** <em style="">f</em> is of at most polynomial growth at cusps of SL<sub style="line-height: 1em;">2</sub>('''Z'''). | ||
| + | |||
| + | |||
| + | ==two types of Maass forms== | ||
| + | * square integrable Maass forms ~ discrete spectrum | ||
| + | * Eisenstein series ~ continuous spectrum | ||
| + | |||
| + | |||
| + | ==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} | ||
| + | </math> | ||
| + | where <math>K_{\nu}</math> is the modified Bessel function of the second kind | ||
| + | * 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) | ||
| + | </math> | ||
| + | |||
| + | |||
| + | ==examples== | ||
| + | ===Eisenstein series=== | ||
* {{수학노트|url=실해석적_아이젠슈타인_급수}} | * {{수학노트|url=실해석적_아이젠슈타인_급수}} | ||
| − | + | ||
| + | ===Maass-Poincare series=== | ||
| + | * Hejhal | ||
| + | * real analytic eigenfunction of the Laplacian with known singularities at <math>i\infty</math> | ||
| + | |||
| + | |||
==Kloosterman sum== | ==Kloosterman sum== | ||
* {{수학노트|url=클루스터만_합}} | * {{수학노트|url=클루스터만_합}} | ||
| − | + | ||
==related items== | ==related items== | ||
| 23번째 줄: | 53번째 줄: | ||
* [[harmonic Maass forms]] | * [[harmonic Maass forms]] | ||
* [[spectral theory of automorphic forms]] | * [[spectral theory of automorphic forms]] | ||
| − | + | * [[q-series and Maass forms]] | |
| − | + | ||
==books== | ==books== | ||
* 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]]) | ||
| − | + | ||
| + | ==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== | ==encyclopedia== | ||
| 46번째 줄: | 82번째 줄: | ||
* http://mathoverflow.net/questions/52744/what-is-the-relationship-between-modular-forms-and-maass-forms | * http://mathoverflow.net/questions/52744/what-is-the-relationship-between-modular-forms-and-maass-forms | ||
| + | |||
| + | |||
| + | |||
| + | ==expositions== | ||
| + | * Jianya Liu [http://www.prime.sdu.edu.cn/lectures/LiuMaassforms.pdf 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. | ||
| 52번째 줄: | 99번째 줄: | ||
[[분류:math]] | [[분류:math]] | ||
[[분류:automorphic forms]] | [[분류:automorphic forms]] | ||
| + | [[분류:migrate]] | ||
| + | |||
| + | ==메타데이터== | ||
| + | ===위키데이터=== | ||
| + | * ID : [https://www.wikidata.org/wiki/Q6721246 Q6721246] | ||
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LOWER': 'maass'}, {'LOWER': 'wave'}, {'LEMMA': 'form'}] | ||
| + | * [{'LOWER': 'maass'}, {'LOWER': 'cusp'}, {'LEMMA': 'form'}] | ||
| + | * [{'LOWER': 'maass'}, {'LEMMA': 'form'}] | ||
2021년 2월 17일 (수) 01:22 기준 최신판
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.
메타데이터
위키데이터
- ID : Q6721246
Spacy 패턴 목록
- [{'LOWER': 'maass'}, {'LOWER': 'wave'}, {'LEMMA': 'form'}]
- [{'LOWER': 'maass'}, {'LOWER': 'cusp'}, {'LEMMA': 'form'}]
- [{'LOWER': 'maass'}, {'LEMMA': 'form'}]