"타원 모듈라 j-함수 (elliptic modular function, j-invariant)"의 두 판 사이의 차이

2012년 6월 13일 (수) 16:51 판

\(j(\tau)= {E_4(\tau)^3\over \Delta(\tau)}=\frac{(1+240\sum_{n>0}\sigma_3(n)q^n)^3}{q-24q+252q^2+\cdots} =q^{-1}+744+196884q+21493760q^2+\cdots\)

여기서 \(q=e^{2\pi i\tau}\)

\( E_4(\tau)=1+240\sum_{n>0}\sigma_3(n)q^n= 1+240q+2160q^2+\cdots\) (아이젠슈타인 급수(Eisenstein series))

\((\sigma_3(n)=\sum_{d|n}d^3)\)

\(\Delta(\tau)= q\prod_{n>0}(1-q^n)^{24}= q-24q+252q^2+\cdots\)

\(j(\tau)=1728\frac{g_2^3}{g_2^3-27g_3^2}\)

1, 744, 196884, 21493760, 864299970, 20245856256, 333202640600, 4252023300096, 44656994071935, 401490886656000, 3176440229784420, 22567393309593600, 146211911499519294, 874313719685775360, 4872010111798142520, 25497827389410525184
[1]http://www.research.att.com/~njas/sequences/A000521
근사식 [Rademacher1938]
\(c(n)=\frac{2\pi}{\sqrt{n}}\sum_{k=1}^\infty \frac{A_k(n)}{k}I_{1}(\frac{4\pi\sqrt{n}}{k})\)
\(I_1\) 은 베셀함수
\(A_k(n)=\sum_{0 \leq h < k,(h,k)=1}e^{-2\pi i(nh+h')/k }\)\(hh'\equiv -1 \mod k\)

복소상반평면의 대수적 수 \(\tau\in \mathbb{H} \cap \bar{\mathbb{Q}}\)에 대하여, \(\tau\)가 2차(imaginary quadratic)가 아니면, \(j(\tau)\)는 초월수이다.

Hilbert class fields of imaginary quadratic fields are generated by singular moduli
확인필요

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-<h5>관련도서 및 추천도서</h5>
+<h5>관련도서</h5>
 * Ranestad, Kristian, ed. 2008. The 1-2-3 of Modular Forms. Berlin, Heidelberg: Springer Berlin Heidelberg. http://www.springerlink.com/content/r351n3u4608rm816/.
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