Non-holomorphic modular forms
imported>Pythagoras0님의 2012년 10월 28일 (일) 15:58 판
weight 2 Eisenstein series==
- \(k=1\)인 경우의 아이젠슈타인급수는 위에서 얻은 푸리에 급수를 이용하여 정의
\(G_{2}(\tau) = \zeta(2) \left(1-24\sum_{n=1}^{\infty} \sigma_{1}(n)q^{n} \right)\)
- 원래의 정의와 비슷하게 쓰려면 절대수렴하지 않는 급수 다음과 같이 덧셈의 순서를 따름
\(G_{2}(\tau) = \frac{1}{2}\sum_{n\neq 0} \frac{1}{n^2}+\frac{1}{2}\sum_{m\neq0}\sum_{n\in\mathbb{Z}} \frac{1}{(m\tau+n)^{2}}\)
- 정규 아이젠슈타인 급수
\(E_{2}(\tau) = 1-24\sum_{n=1}^{\infty} \sigma_{1}(n)q^{n}\)
- modularity
\(G_{2} \left( \frac{ a\tau +b}{ c\tau + d} \right) = (c\tau +d)^{2} G_{2}(\tau)-\pi i c(c\tau+d)\)
E2 as a non-holomorphic modular form==
- \(\tau = x+ iy\), \(y > 0 \)에 대하여 다음과 정의된 함수는 모듈라 성질을 가짐
\(G^{*}_{2}(\tau) = G_{2}(\tau)-\frac{\pi}{2y}\)
\(E^{*}_{2}(\tau) = E_{2}(\tau)-\frac{3}{\pi y}\)
- obtaing modularity losing holomorphicity
Zagier's function
- Hurwitz class numbers
- Cox_on_Hurwitz_class_number.pdf (Cox's book 319p)
- Zagier's paper
- Zagier-Hirzebruch function
- Intersection numbers of curves on Hibert modular surfaces and modular forms of Nebentypus
- function with coefficients as Hurwitz class numbers
- zagier_hirzebruch.pdf
- Sums involving the values at negative integers of L-functions of quadratic characters
- Henri Cohen, 1975
\(G_{2}(\tau) = \zeta(2) \left(1-24\sum_{n=1}^{\infty} \sigma_{1}(n)q^{n} \right)\)
\(G_{2}(\tau) = \frac{1}{2}\sum_{n\neq 0} \frac{1}{n^2}+\frac{1}{2}\sum_{m\neq0}\sum_{n\in\mathbb{Z}} \frac{1}{(m\tau+n)^{2}}\)
\(E_{2}(\tau) = 1-24\sum_{n=1}^{\infty} \sigma_{1}(n)q^{n}\)
\(G_{2} \left( \frac{ a\tau +b}{ c\tau + d} \right) = (c\tau +d)^{2} G_{2}(\tau)-\pi i c(c\tau+d)\)
- \(\tau = x+ iy\), \(y > 0 \)에 대하여 다음과 정의된 함수는 모듈라 성질을 가짐
\(G^{*}_{2}(\tau) = G_{2}(\tau)-\frac{\pi}{2y}\)
\(E^{*}_{2}(\tau) = E_{2}(\tau)-\frac{3}{\pi y}\) - obtaing modularity losing holomorphicity
- Intersection numbers of curves on Hibert modular surfaces and modular forms of Nebentypus
- function with coefficients as Hurwitz class numbers
- zagier_hirzebruch.pdf
- Henri Cohen, 1975