"Maass forms"의 두 판 사이의 차이

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* Hyperbolic distribution problems and half-integral weight Maass forms
 
* Hyperbolic distribution problems and half-integral weight Maass forms
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* 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.
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* Automorphic forms correspond to representations that occur in <math>L_2(\Gamma\backslash G)</math>.  
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* In the case when <math>G</math> is <math>SL(2,\mathbb{R})</math>
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** holomorphic modular forms correspond to (highest weight vectors of) discrete series representations of <math>G</math>
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** Maass wave forms correspond to (spherical vectors of) continuous series representations of G.
  
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==definition==
  
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==Eisenstein series==
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* 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:
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** <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.
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** <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>
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** <em style="">f</em> is of at most polynomial growth at cusps of SL<sub style="line-height: 1em;">2</sub>('''Z''').
  
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* z = x + iy in the upper half-plane
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* Re(s) > 1
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==two types of Maass forms==
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*  definition<br><math>E(z,s) ={1\over 2}\sum_{(m,n)=1}{y^s\over|mz+n|^{2s}}</math><br>
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* square integrable Maass forms ~ discrete spectrum
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*  Maass form<br><math>\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)</math><br>
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* Eisenstein series ~ continuous spectrum
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* functional equation<br><math>E^{*}(z,s) = \pi^{-s}\Gamma(s)\zeta(2s)E(z,s) </math><br><math>E^{*}(z,s)=E^{*}(z,1-s)</math><br>
 
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* a unique pole of residue 3/π at s = 1
 
  
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==fourier expansion==
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* <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
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:<math>
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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}
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</math>
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where <math>K_{\nu}</math> is the modified Bessel function of the second kind
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* under the assumption that <math>f(x+iy)=f(-x+iy)</math>, we get
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:<math>
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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)
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</math>
  
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==Kloosterman sum==
 
  
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* used to estimate the Fourier coefficients of modular forms
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==examples==
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*  definition for prime p<br><math>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</math><br>
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===Eisenstein series===
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* generally defined as<br><math>K(a,b;m)=\sum_{0\leq x\leq m-1,\ gcd(x,m)=1 } e^{2\pi i (ax+bx^*)/m}</math><br>
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* {{수학노트|url=실해석적_아이젠슈타인_급수}}
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* http://blogs.ethz.ch/kowalski/2010/02/26/the-fourth-moment-of-kloosterman-sums/
 
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* Kloosterman, H. D. [http://www.springerlink.com/content/cq7681384842j128/?p=5679d1bb49fd45a3987db6d83a1147b6&pi=1 On the representation of numbers in the form ax² + by² + cz² + dt²], Acta Mathematica 49 (1926), pp. 407-464
 
  
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===Maass-Poincare series===
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* Hejhal
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* real analytic eigenfunction of the Laplacian with known singularities at <math>i\infty</math>
  
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==history==
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
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==Kloosterman sum==
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* {{수학노트|url=클루스터만_합}}
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==related items==
 
==related items==
  
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* [[harmonic Maass forms|examples of harmonic Maass Forms]]
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* [[harmonic Maass forms]]
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* [[spectral theory of automorphic forms]]
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* [[q-series and Maass forms]]  
  
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==books==
 
==books==
  
 
* Henryk Iwaniek, Emmanuel Kowalski (2004). Analytic number theory
 
* Henryk Iwaniek, Emmanuel Kowalski (2004). Analytic number theory
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*  Lectures on modular functions of one complex variable (Tata Institute of Fundamental Research. Lectures on mathematics and physics. Mathematics, 29) <br>
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*  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]])
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* [[4909919|찾아볼 수학책]]<br>
 
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* http://gigapedia.info/1/Iwaniek
 
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* http://gigapedia.info/1/Maass
 
  
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==computational resource==
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* https://docs.google.com/file/d/0B8XXo8Tve1cxMll2Y2lzeWg0UVE/edit
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* http://www.lmfdb.org/ModularForm/GL2/Q/Maass/
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* http://www.math.chalmers.se/~sj/forskning/level11_2a.pdf
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* http://www.ijpam.eu/contents/2009-54-2/12/12.pdf
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* https://mathoverflow.net/questions/22908/does-anyone-want-a-pretty-maass-form
  
 
==encyclopedia==
 
==encyclopedia==
  
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* [http://en.wikipedia.org/wiki/Kronecker_limit_formula ]http://en.wikipedia.org/wiki/Kronecker_limit_formula
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* http://en.wikipedia.org/wiki/Kronecker_limit_formula
 
* http://en.wikipedia.org/wiki/Real_analytic_Eisenstein_series
 
* http://en.wikipedia.org/wiki/Real_analytic_Eisenstein_series
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* http://en.wikipedia.org/wiki/Kloosterman_sum
 
  
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==question and answers(Math Overflow)==
 
==question and answers(Math Overflow)==
  
 
* 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
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* http://mathoverflow.net/search?q=
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[[분류:개인노트]][[분류:개인노트]]
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[[분류:math and physics]]
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==expositions==
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* Jianya Liu [http://www.prime.sdu.edu.cn/lectures/LiuMaassforms.pdf LECTURES ON MAASS FORMS]
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==articles==
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* 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
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* 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.
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* 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.
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[[분류:개인노트]]
 
[[분류:math and physics]]
 
[[분류:math and physics]]
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[[분류:math]]
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[[분류:automorphic forms]]
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[[분류:migrate]]
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==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q6721246 Q6721246]
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===Spacy 패턴 목록===
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* [{'LOWER': 'maass'}, {'LOWER': 'wave'}, {'LEMMA': 'form'}]
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* [{'LOWER': 'maass'}, {'LOWER': 'cusp'}, {'LEMMA': 'form'}]
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* [{'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


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)


expositions


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

Spacy 패턴 목록

  • [{'LOWER': 'maass'}, {'LOWER': 'wave'}, {'LEMMA': 'form'}]
  • [{'LOWER': 'maass'}, {'LOWER': 'cusp'}, {'LEMMA': 'form'}]
  • [{'LOWER': 'maass'}, {'LEMMA': 'form'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=Maass_forms&oldid=51733"