"타원곡선"의 두 판 사이의 차이
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| 38번째 줄: | 38번째 줄: | ||
* the only possible torsion groups for elliptic curves over Q are the cyclic groups of order 1,2,3,4,5,6,7,8,9,10,12 [sic -- 11 is not possible] and<br><math>\frac{\mathbb Z}{2\mathbb Z}\oplus \frac{\mathbb Z}{n\mathbb Z}</math> for n=1,2,3,4<br> | * the only possible torsion groups for elliptic curves over Q are the cyclic groups of order 1,2,3,4,5,6,7,8,9,10,12 [sic -- 11 is not possible] and<br><math>\frac{\mathbb Z}{2\mathbb Z}\oplus \frac{\mathbb Z}{n\mathbb Z}</math> for n=1,2,3,4<br> | ||
| − | * <math>y^2=x^3-n^2x</math> | + | * 예) <math>E_n : y^2=x^3-n^2x</math>의 torsion은 <math>\{(\infty,\infty), (0,0),(n,0),(-n,0)\}</math>임 |
| 112번째 줄: | 112번째 줄: | ||
* [http://ko.wikipedia.org/wiki/%ED%83%80%EC%9B%90%EA%B3%A1%EC%84%A0 http://ko.wikipedia.org/wiki/타원곡선] | * [http://ko.wikipedia.org/wiki/%ED%83%80%EC%9B%90%EA%B3%A1%EC%84%A0 http://ko.wikipedia.org/wiki/타원곡선] | ||
| − | * http://en.wikipedia.org/wiki/elliptic_curve | + | * [http://en.wikipedia.org/wiki/elliptic_curve ]http://en.wikipedia.org/wiki/elliptic_curve |
| + | * http://en.wikipedia.org/wiki/Mordell-Weil_theorem | ||
* http://en.wikipedia.org/wiki/ | * http://en.wikipedia.org/wiki/ | ||
* [http://www.wolframalpha.com/input/?i=y%5E2=x%5E3-x http://www.wolframalpha.com/input/?i=y^2=x^3-x] | * [http://www.wolframalpha.com/input/?i=y%5E2=x%5E3-x http://www.wolframalpha.com/input/?i=y^2=x^3-x] | ||
| 151번째 줄: | 152번째 줄: | ||
** Silverman, Joseph H. (1986), Graduate Texts in Mathematics, 106, Springer-Verlag<br> | ** Silverman, Joseph H. (1986), Graduate Texts in Mathematics, 106, Springer-Verlag<br> | ||
* 도서내검색<br> | * 도서내검색<br> | ||
| − | ** http://books.google.com/books?q= | + | ** [http://books.google.com/books?q=%ED%83%80%EC%9B%90%EA%B3%A1%EC%84%A0 http://books.google.com/books?q=타원곡선] |
** [http://book.daum.net/search/contentSearch.do?query=%ED%83%80%EC%9B%90%EA%B3%A1%EC%84%A0 http://book.daum.net/search/contentSearch.do?query=타원곡선] | ** [http://book.daum.net/search/contentSearch.do?query=%ED%83%80%EC%9B%90%EA%B3%A1%EC%84%A0 http://book.daum.net/search/contentSearch.do?query=타원곡선] | ||
** http://book.daum.net/search/contentSearch.do?query= | ** http://book.daum.net/search/contentSearch.do?query= | ||
2009년 10월 14일 (수) 22:43 판
간단한 소개
격자와 타원곡선
\(y^2=4x^3-g_2(\tau)x-g_3\)
\(g_2(\tau) = 60G_4=60\sum_{ (m,n) \neq (0,0)} \frac{1}{(m+n\tau )^{4}}\)
\(g_3(\tau) = 140G_6=140\sum_{ (m,n) \neq (0,0)} \frac{1}{(m+n\tau )^{6}}\)
군의 구조
- chord-tangent method
- 유리수해에 대한 Mordell-Weil theorem
rank와 torsion
- the only possible torsion groups for elliptic curves over Q are the cyclic groups of order 1,2,3,4,5,6,7,8,9,10,12 [sic -- 11 is not possible] and
\(\frac{\mathbb Z}{2\mathbb Z}\oplus \frac{\mathbb Z}{n\mathbb Z}\) for n=1,2,3,4 - 예) \(E_n : y^2=x^3-n^2x\)의 torsion은 \(\{(\infty,\infty), (0,0),(n,0),(-n,0)\}\)임
예
- \(y^2=x^3-x\)
[/pages/2061314/attachments/2299029 MSP1975197gdf732cih44i50000361d01gd578fhc4a.gif] - 유리수해
\(E(\mathbb Q)=\{(\infty,\infty), (0,0),(1,0),(-1,0)\} \simeq \frac{\mathbb Z}{2\mathbb Z}\oplus \frac{\mathbb Z}{2\mathbb Z}\) - 주기
\(2\omega=4\int_0^1\frac{dx}{\sqrt{1-x^4}}=B(1/2,1/4)=\frac{\Gamma(\frac{1}{2})\Gamma(\frac{1}{4})}{\Gamma(\frac{3}{4})}=5.24\cdots\)
L-함수
타니야마-시무라 추측(정리)
Birch and Swinnerton-Dyer conjecture
재미있는 사실
역사
관련된 다른 주제들
- 타원적분
- lemniscate 곡선의 길이와 타원적분
- 정수계수 이변수 이차형식(binary integral quadratic forms)
- j-invariant
- 아이젠슈타인 급수(Eisenstein series)
- 베타적분
- 사각 피라미드 퍼즐
- congruent number 문제
수학용어번역
사전 형태의 자료
- http://ko.wikipedia.org/wiki/타원곡선
- [1]http://en.wikipedia.org/wiki/elliptic_curve
- http://en.wikipedia.org/wiki/Mordell-Weil_theorem
- http://en.wikipedia.org/wiki/
- http://www.wolframalpha.com/input/?i=y^2=x^3-x
- http://www.wolframalpha.com/input/?i=
- NIST Digital Library of Mathematical Functions
관련논문
- Conics - a Poor Man's Elliptic Curves
- Franz Lemmermeyer, arXiv:math/0311306v1
- Elliptic Curves
- John Stillwell, The American Mathematical Monthly, Vol. 102, No. 9 (Nov., 1995), pp. 831-837
- Three Fermat Trails to Elliptic Curves
- Ezra Brown, The College Mathematics Journal, Vol. 31, No. 3 (May, 2000), pp. 162-172
- Rational isogenies of prime degree
- Barry Mazur, Inventiones Math. 44 (1978), 129--162
- http://www.jstor.org/action/doBasicSearch?Query=elliptic+curves
- http://www.jstor.org/action/doBasicSearch?Query=congruent+number+problem
- http://www.jstor.org/action/doBasicSearch?Query=
관련도서 및 추천도서
- Introduction to elliptic curves and modular forms
- Neal Koblitz - 1993
- Rational points on elliptic curves
- Joseph H. Silverman, John Torrence Tate - 1992
- 학부생의 입문용으로 좋은 책
- The Arithmetic of Elliptic Curves
- Silverman, Joseph H. (1986), Graduate Texts in Mathematics, 106, Springer-Verlag
- Silverman, Joseph H. (1986), Graduate Texts in Mathematics, 106, Springer-Verlag
- 도서내검색
- 도서검색
관련기사
- 네이버 뉴스 검색 (키워드 수정)