타원 모듈라 j-함수의 singular moduli

개요

• 타원 모듈라 j-함수의 quadratic imaginary number 에서의 값들
• 중요한 결과로 Gross-Zagier 공식이 있음

예

\begin{array}{c|c|c|c} \tau & j(\tau) & \sqrt[3]{j(\tau )} & \text{factorization} \\ \hline i & 1728 & 12 & 2^6\cdot 3^3 \\ \frac{1}{2} \left(-1+i \sqrt{3}\right) & 0 & 0 & 0 \\ \frac{1}{2} \left(-1+i \sqrt{7}\right) & -3375 & -15 & -3^3 5^3 \\ i \sqrt{2} & 8000 & 20 & 2^6\cdot 5^3 \\ \frac{1}{2} \left(-1+i \sqrt{11}\right) & -32768 & -32 & -2^{15} \\ \frac{1}{2} \left(-1+i \sqrt{19}\right) & -884736 & -96 & -2^{15} 3^3 \\ \frac{1}{2} \left(-1+i \sqrt{43}\right) & -884736000 & -960 & -2^{18} 3^3 5^3 \\ \frac{1}{2} \left(-1+i \sqrt{67}\right) & -147197952000 & -5280 & -2^{15} 3^3 5^3 11^3 \\ \frac{1}{2} \left(-1+i \sqrt{163}\right) & -262537412640768000 & -640320 & -2^{18} 3^3 5^3 23^3 29^3 \\ i \sqrt{3} & 54000 & 30 \sqrt[3]{2} & 2^4\cdot 3^3\cdot 5^3 \\ 2 i & 287496 & 66 & 2^3\cdot 3^3\cdot 11^3 \\ i \sqrt{7} & 16581375 & 255 & 3^3\cdot 5^3\cdot 17^3 \\ \frac{1}{2} \left(-1+3 i \sqrt{3}\right) & -12288000 & 160 \sqrt[3]{-3} & -2^{15} 3^1 5^3 \\ \sqrt{-5} & 632000+282880 \sqrt{5} & 50+26\sqrt{5} & \\ \frac {-1+\sqrt{-5}}{2} & 632000-282880 \sqrt{5} & 50-26\sqrt{5} & \end{array}

Gross-Zagier 공식

• \(d_1, d_ 2\)가 서로 다른 두 복소이차수체 \(K_1, K_2\)의 판별식이라 하자.
• \(J(d_ 1,d_ 2)\)를 \(\prod_{}\left(j(\alpha_1)-j(\alpha_2)\right)^{\frac{4}{w_1 w_2}}\) 로 정의하자. 여기서 \(\alpha_1, \alpha_2\) 는 각각 \(K_1\), \(K_2\) 의 ideal class의 representatives

정리 [Gross-Zagier]

\[J (d_ 1,d_ 2)^2=\prod_{\substack{x,n,n'\in \mathbb{Z}, \\ x^2+4nn'=d_ 1d_ 2, \\ n,n'>0}}n^{\epsilon(n')}\]

역사

• http://www.google.com/search?hl=en&tbs=tl:1&q=
• 수학사 연표

메모

• Morton, Patrick. “Genus Theory and the Factorization of Class Equations over \(\mathbb{F}_p\).” arXiv:1409.0691 [math], September 2, 2014. http://arxiv.org/abs/1409.0691.
• http://terrytao.wordpress.com/2007/05/03/distinguished-lecture-series-ii-shou-wu-zhang-“gross-zagier-formula-and-birch-and-swinnerton-dyer-conjecture-”/
• asymptotics http://mathoverflow.net/questions/48544/bounding-the-modular-discriminant-of-an-elliptic-curve-in-the-j-invariant

관련된 항목들

• complex multiplication
• 정수계수 이변수 이차형식(binary integral quadratic forms)

수학용어번역

• singular - 대한수학회 수학용어집
  • 특이
• moduli - 대한수학회 수학용어집

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxU3MtUWtoTS1kaW8/edit

리뷰, 에세이, 강의노트

• Duke and Jenkins, Notes on singular moduli and modular forms

관련논문

• Ehlen, Stephan. ‘Singular Moduli of Higher Level and Special Cycles’. arXiv:1505.02711 [math], 11 May 2015. http://arxiv.org/abs/1505.02711.
• Bilu, Yuri, Florian Luca, and Amalia Pizarro-Madariaga. “Rational Products of Singular Moduli.” arXiv:1410.1806 [math], October 7, 2014. http://arxiv.org/abs/1410.1806.
• Lauter, Kristin, and Bianca Viray. “On Singular Moduli for Arbitrary Discriminants.” International Mathematics Research Notices, December 1, 2014. doi:10.1093/imrn/rnu223.
• Howard, Benjamin, and Tonghai Yang. 2012. “Singular Moduli Refined.” arXiv:1202.6410 (February 28). http://arxiv.org/abs/1202.6410.
• Dorman, David R. “Special values of the elliptic modular function and factorization formulae.” Journal für die reine und angewandte Mathematik 383 (1988): 207–20.
• On singular moduli. Gross, B.H.; Zagier, Don B, J. Rcinc Angew. Math. 355 (1984), 191-220

• Birkoff  a sourcebook in classical analysis
• SL(2,Z) invariant spaces spanned by modular units
• examples http://w3.countnumber.de/addons/Publikationen/text/mex.pdf
• Explicit elliptic units, I
  • Farshid Hajir and Fernando Rodriguez Villegas
  • Source: Duke Math. J. Volume 90, Number 3 (1997), 495-521.
• Gross, B.H.; Zagier, Don B On singular moduli.
• Ken Ono Singular moduli generating functions for modular curves and surfaces.

Singular moduli, modular polynomials, and the index of the closure of Z[j(

• • D. Dorman, Math. Ann. 283 (1989), pages 177-191.
• Special values of the elliptic modular function and factorization formulae
  • D. Dorman,  J. reine angew. Math. 383 (1988), pages 207-220