"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 | ||
| − | * Automorphic forms correspond to representations that occur in | + | * Automorphic forms correspond to representations that occur in $L_2(G/\Gamma)$. In the case when $G$ is $SL_2$, 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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* used to estimate the Fourier coefficients of modular forms | * used to estimate the Fourier coefficients of modular forms | ||
| − | * definition for prime p | + | * definition for prime p |
| − | * generally defined as | + | ;<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 | ||
| + | :<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/ | * 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 | * 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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==related items== | ==related items== | ||
| − | * [[harmonic Maass forms | + | * [[harmonic Maass forms]] |
| + | * [[automorphic forms]] | ||
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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> | * Lectures on modular functions of one complex variable (Tata Institute of Fundamental Research. Lectures on mathematics and physics. Mathematics, 29) <br> | ||
** Hans Maass, ([[5323613/attachments/3133467|pdf]]) | ** Hans Maass, ([[5323613/attachments/3133467|pdf]]) | ||
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* 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 | ||
* http://mathoverflow.net/search?q= | * http://mathoverflow.net/search?q= | ||
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[[분류:개인노트]] | [[분류:개인노트]] | ||
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[[분류:math and physics]] | [[분류:math and physics]] | ||
[[분류:math]] | [[분류:math]] | ||
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2013년 3월 17일 (일) 12:42 판
introduction
- Hyperbolic distribution problems and half-integral weight Maass forms
- Automorphic forms correspond to representations that occur in $L_2(G/\Gamma)$. In the case when $G$ is $SL_2$, 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.
Eisenstein series
Kloosterman sum
- used to estimate the Fourier coefficients of modular forms
- definition for prime p
- \(S(a,b;p)=\sum_{1\leq x\leq p-1}{\exp(2i\pi (ax+b\bar{x})/p)},\quad\text{where}\quad x\bar{x}\equiv 1\text{ mod } p\)
- generally defined as
\[K(a,b;m)=\sum_{0\leq x\leq m-1,\ gcd(x,m)=1 } e^{2\pi i (ax+bx^*)/m}\]
- http://blogs.ethz.ch/kowalski/2010/02/26/the-fourth-moment-of-kloosterman-sums/
- Kloosterman, H. D. On the representation of numbers in the form ax² + by² + cz² + dt², Acta Mathematica 49 (1926), pp. 407-464
history
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)
encyclopedia
- http://en.wikipedia.org/wiki/Kronecker_limit_formula
- http://en.wikipedia.org/wiki/Real_analytic_Eisenstein_series
- http://en.wikipedia.org/wiki/Kloosterman_sum