"Maass forms"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
 
(사용자 3명의 중간 판 36개는 보이지 않습니다)
1번째 줄: 1번째 줄:
−
<h5>introduction</h5>
+
==introduction==
  
 
* 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 <math>L_2(\Gamma\backslash G)</math>.
 +
* 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 <math>G</math>
 +
** Maass wave forms correspond to (spherical vectors of) continuous series representations of G.
  
−
 
+
 +
==definition==
  
−
 
+
* 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''').
  
−
<h5>Eisenstein series</h5>
+
 +
==two types of Maass forms==
 +
* square integrable Maass forms ~ discrete spectrum
 +
* Eisenstein series ~ continuous spectrum
  
−
* z = x + iy in the upper half-plane
 
−
* Re(s) > 1
 
−
*  definition<br><math>E(z,s) ={1\over 2}\sum_{(m,n)=1}{y^s\over|mz+n|^{2s}}</math><br>
 
−
*  Maass form<br><math>\DeltaE(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>
 
−
*  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>
 
−
* a unique pole of residue 3/π at s = 1
 
  
−
 
+
==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>
  
−
 
 
  
−
<h5>Kloosterman sum</h5>
+
==examples==
 +
===Eisenstein series===
 +
* {{수학노트|url=실해석적_아이젠슈타인_급수}}
  
−
* used to estimate the Fourier coefficients of modular forms
 
−
*  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>
 
−
*  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>
 
−
* http://blogs.ethz.ch/kowalski/2010/02/26/the-fourth-moment-of-kloosterman-sums/
 
−
* 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
 
−
*  
 
  
−
 
+
===Maass-Poincare series===
 +
* Hejhal
 +
* real analytic eigenfunction of the Laplacian with known singularities at <math>i\infty</math>
  
−
 
+
  
−
<h5>history</h5>
+
==Kloosterman sum==
 +
* {{수학노트|url=클루스터만_합}}
  
−
* http://www.google.com/search?hl=en&tbs=tl:1&q=
+
  
−
 
+
==related items==
  
−
 
+
* [[harmonic Maass forms]]
 +
* [[spectral theory of automorphic forms]]
 +
* [[q-series and Maass forms]]
  
−
<h5>related items</h5>
+
  
−
* [[harmonic Maass forms|examples of harmonic 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, ([[5323613/attachments/3133467|pdf]])
−
 
 
−
<h5>books</h5>
 
  
−
* Henryk Iwaniek, Emmanuel Kowalski (2004). Analytic number theory
 
−
* [[4909919|찾아볼 수학책]]<br>
 
−
* http://gigapedia.info/1/Iwaniek
 
−
* http://gigapedia.info/1/
 
−
* http://gigapedia.info/1/
 
−
* http://gigapedia.info/1/
 
−
* http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
 
  
−
 
 
  
−
 
+
==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
  
−
<h5>encyclopedia</h5>
+
==encyclopedia==
  
−
* http://ko.wikipedia.org/wiki/
+
* http://en.wikipedia.org/wiki/Kronecker_limit_formula
−
* [http://en.wikipedia.org/wiki/Kronecker_limit_formula ]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
−
* http://en.wikipedia.org/wiki/Kloosterman_sum
 
−
* http://en.wikipedia.org/wiki/
 
−
* http://en.wikipedia.org/wiki/
 
−
* http://en.wikipedia.org/wiki/
 
−
* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
 
−
 
−
 
 
−
 
−
 
 
−
 
−
<h5>question and answers(Math Overflow)</h5>
 
−
 
−
* http://mathoverflow.net/search?q=
 
−
* http://mathoverflow.net/search?q=
 
−
* http://mathoverflow.net/search?q=
 
−
 
−
 
 
−
 
−
 
 
−
 
−
<h5>blogs</h5>
 
−
 
−
*  구글 블로그 검색<br>
 
−
** http://blogsearch.google.com/blogsearch?q=
 
−
** http://blogsearch.google.com/blogsearch?q=
 
−
** http://blogsearch.google.com/blogsearch?q=
 
  
−
 
 
  
−
 
 
  
−
<h5>articles</h5>
+
==question and answers(Math Overflow)==
  
−
* [[2010년 books and articles|논문정리]]
+
* http://mathoverflow.net/questions/52744/what-is-the-relationship-between-modular-forms-and-maass-forms
−
* http://www.ams.org/mathscinet
 
−
* http://www.zentralblatt-math.org/zmath/en/
 
−
* http://pythagoras0.springnote.com/
 
−
* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html]
 
−
* http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
 
−
* http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
 
−
* http://dx.doi.org/
 
  
−
 
 
  
−
 
 
  
−
<h5>experts on the field</h5>
+
==expositions==
 +
* Jianya Liu [http://www.prime.sdu.edu.cn/lectures/LiuMaassforms.pdf LECTURES ON MAASS FORMS]
  
−
* http://arxiv.org/
 
  
−
 
+
==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.
  
−
 
 
  
−
<h5>TeX </h5>
+
[[분류:개인노트]]
 +
[[분류:math and physics]]
 +
[[분류:math]]
 +
[[분류:automorphic forms]]
 +
[[분류:migrate]]
  
−
* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
+
==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


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"