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2013년 3월 17일 (일) 13:42 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