"Maass forms"의 두 판 사이의 차이

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

틀:수학노트

Maass-Poincare series

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

Kloosterman sum

틀:수학노트

related items

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

encyclopedia

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'}]

@@ 2번째 줄: / 2번째 줄: @@
 * Hyperbolic distribution problems and half-integral weight Maass forms
-* Automorphic forms correspond to representations that occur in $L_2(\Gamma\backslash G)$.
+* Automorphic forms correspond to representations that occur in <math>L_2(\Gamma\backslash G)</math>.
-* In the case when $G$ is $SL_2$
+* 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 $G$
+** 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.
+==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}}
+===Maass-Poincare series===
+* Hejhal
+* real analytic eigenfunction of the Laplacian with known singularities at <math>i\infty</math>
 ==Kloosterman sum==
 * {{수학노트|url=클루스터만_합}}
 ==related items==
@@ 23번째 줄: / 53번째 줄: @@
 * [[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) <br>
+*  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]])
-==expositions==
-* Jianya Liu [http://www.prime.sdu.edu.cn/lectures/LiuMaassforms.pdf LECTURES ON MAASS FORMS]
 ==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==
@@ 55번째 줄: / 82번째 줄: @@
 * 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.
@@ 61번째 줄: / 99번째 줄: @@
 [[분류:math]]
 [[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'}]