"타원 모듈라 j-함수의 singular moduli"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) (→관련논문) |
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| (사용자 2명의 중간 판 25개는 보이지 않습니다) | |||
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| − | + | ==개요== | |
| − | * [[타원 모듈라 j- | + | * [[타원 모듈라 j-함수 (elliptic modular function, j-invariant)|타원 모듈라 j-함수]]의 quadratic imaginary number 에서의 값들 |
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* 중요한 결과로 Gross-Zagier 공식이 있음 | * 중요한 결과로 Gross-Zagier 공식이 있음 | ||
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| − | + | ==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} | ||
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| − | + | ==Gross-Zagier 공식== | |
| − | + | * <math>d_1, d_ 2</math>가 서로 다른 두 복소이차수체 <math>K_1, K_2</math>의 판별식이라 하자. | |
| + | * <math>J(d_ 1,d_ 2)</math>를 <math>\prod_{}\left(j(\alpha_1)-j(\alpha_2)\right)^{\frac{4}{w_1 w_2}}</math> 로 정의하자. 여기서 <math>\alpha_1, \alpha_2</math> 는 각각 <math>K_1</math>, <math>K_2</math> 의 ideal class의 representatives | ||
| + | ;정리 [Gross-Zagier] | ||
| + | :<math>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')}</math> | ||
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| − | + | ==역사== | |
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* http://www.google.com/search?hl=en&tbs=tl:1&q= | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
| − | * [[ | + | * [[수학사 연표]] |
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| − | < | + | ==메모== |
| + | * Morton, Patrick. “Genus Theory and the Factorization of Class Equations over <math>\mathbb{F}_p</math>.” 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-%E2%80%9Cgross-zagier-formula-and-birch-and-swinnerton-dyer-conjecture-%E2%80%9D/ 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)]] | ||
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| − | + | ==수학용어번역== | |
| + | * {{학술용어집|url=singular}} | ||
| + | ** 특이 | ||
| + | * {{학술용어집|url=moduli}} | ||
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| − | + | ==매스매티카 파일 및 계산 리소스== | |
* https://docs.google.com/file/d/0B8XXo8Tve1cxU3MtUWtoTS1kaW8/edit | * https://docs.google.com/file/d/0B8XXo8Tve1cxU3MtUWtoTS1kaW8/edit | ||
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| − | + | ==리뷰, 에세이, 강의노트== | |
| + | * Duke and Jenkins, [http://www.maths.bris.ac.uk/~mamjd/bb/notes/duke_jenkins_li_notes.pdf 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. | ||
| + | * [http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=262662 On singular moduli.] Gross, B.H.; Zagier, Don B, J. Rcinc Angew. Math. 355 (1984), 191-220 | ||
| − | + | * Birkoff a sourcebook in classical analysis | |
| + | * [http://dx.doi.org/10.1142/9789812774415_0016 SL(2,Z) invariant spaces spanned by modular units] | ||
| + | * examples http://w3.countnumber.de/addons/Publikationen/text/mex.pdf | ||
| + | * [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1077232812 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 [http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=262662 On singular moduli.] | ||
| + | * Ken Ono [http://www.google.com/url?sa=t&source=web&ct=res&cd=1&url=http%3A%2F%2Fwww.claymath.org%2Fpublications%2FGauss_Dirichlet%2Fono.pdf&ei=e9LGSvXXCYbOsQPMjc2iBQ&usg=AFQjCNHBel_I_9pU-OeXVumQYsCBmouH8A&sig2=syQ4RY4MrY81U0nQyh8BGA Singular moduli generating functions for modular curves and surfaces]. | ||
| + | [http://www.springerlink.com/content/h335012525j22t17/ Singular moduli, modular polynomials, and the index of the closure of Z[j(] | ||
| + | ** D. Dorman, Math. Ann. 283 (1989), pages 177-191. | ||
| + | * [http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN243919689_0383&DMDID=dmdlog9 Special values of the elliptic modular function and factorization formulae] | ||
| + | ** D. Dorman, J. reine angew. Math. 383 (1988), pages 207-220 | ||
| − | + | [[분류:정수론]] | |
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2020년 12월 27일 (일) 23:52 기준 최신판
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
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
수학용어번역
매스매티카 파일 및 계산 리소스
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