"타원곡선"의 두 판 사이의 차이
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* [[타원곡선]]<br> | * [[타원곡선]]<br> | ||
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| 74번째 줄: | 76번째 줄: | ||
<h5>Hasse-Weil 정리</h5> | <h5>Hasse-Weil 정리</h5> | ||
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| + | <math>|\#E(\mathbb{F}_p)-p-1|\leq 2\sqrt{p}</math> | ||
| 86번째 줄: | 90번째 줄: | ||
<h5>타니야마-시무라 추측(정리)</h5> | <h5>타니야마-시무라 추측(정리)</h5> | ||
| − | * [[search?q=%ED%83%80%EB%8B%88%EC%95%BC%EB%A7%88-%EC%8B%9C%EB%AC%B4%EB%9D%BC%20%EC%B6%94%EC%B8%A1 | + | * [[search?q=%ED%83%80%EB%8B%88%EC%95%BC%EB%A7%88-%EC%8B%9C%EB%AC%B4%EB%9D%BC%20%EC%B6%94%EC%B8%A1%28%EC%A0%95%EB%A6%AC%29&parent id=2061314|타니야마-시무라 추측(정리)]] |
| 174번째 줄: | 178번째 줄: | ||
* [http://www.jstor.org/stable/2324954 Taxicabs and Sums of Two Cubes]<br> | * [http://www.jstor.org/stable/2324954 Taxicabs and Sums of Two Cubes]<br> | ||
** Joseph H. SilvermanThe American Mathematical Monthly, Vol. 100, No. 4 (Apr., 1993), pp. 331-340 | ** Joseph H. SilvermanThe American Mathematical Monthly, Vol. 100, No. 4 (Apr., 1993), pp. 331-340 | ||
| + | * [http://www.jstor.org/stable/2690080 Why Study Equations over Finite Fields?]<br> | ||
| + | ** Neal Koblitz, <cite style="line-height: 2em;">Mathematics Magazine</cite>, Vol. 55, No. 3 (May, 1982), pp. 144-149 | ||
2009년 12월 12일 (토) 15:49 판
이 항목의 스프링노트 원문주소
개요
예
- congruent number 문제
방정식 \(y^2=x^3-n^2x\) 이 등장 - 사각 피라미드 퍼즐
\(y^2=\frac{x(x+1)(2x+1)}{6}\)
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 theorem
- 유리수체 위에 정의된 타원의 유리수해는 유한생성아벨군의 구조를 가짐
- \(\mathbb{Z}^r \oplus \mathbb{T}\)
덧셈공식
- \(y^2=x^3+ax^2+bx+c\)위의 점 \(P=(x,y)\)에 대하여,
\(2P\)의 \(x\)좌표는\(\frac{x^4-2bx^2-8cx-4ac+b^2}{4y^2}\) 로 주어진다
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
\(2\omega=4\int_0^1\frac{dx}{\sqrt{1-x^4}}=B(1/2,1/4)=\frac{\Gamma(\frac{1}{2})\Gamma(\frac{1}{4})}{\Gamma(\frac{3}{4})}=\frac{\Gamma(1/4)^2}{\sqrt{2\pi}}=5.24\cdots\)
\(2\int_0^1\frac{dx}{\sqrt{x-x^3}}=B(1/2,1/4)=\frac{\Gamma(\frac{1}{2})\Gamma(\frac{1}{4})}{\Gamma(\frac{3}{4})}=5.24\cdots\) - 모듈라 군, j-invariant and the singular moduli 의 special values 부분과 비교
Hasse-Weil 정리
\(|\#E(\mathbb{F}_p)-p-1|\leq 2\sqrt{p}\)
L-함수
타니야마-시무라 추측(정리)
Birch and Swinnerton-Dyer 추측
재미있는 사실
역사
관련된 다른 주제들
- 타원적분
- periods
- lemniscate 곡선의 길이와 타원적분
- 정수계수 이변수 이차형식(binary integral quadratic forms)
- j-invariant
- 아이젠슈타인 급수(Eisenstein series)
- 베타적분
- 가우스의 class number one 문제
- L-함수, 제타함수와 디리클레 급수
- 무리수와 초월수
수학용어번역
사전 형태의 자료
- http://ko.wikipedia.org/wiki/타원곡선
- http://en.wikipedia.org/wiki/elliptic_curve
- http://en.wikipedia.org/wiki/Mordell-Weil_theorem
- http://en.wikipedia.org/wiki/Heegner_point
- 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
expository articles
- Conics - a Poor Man's Elliptic Curves
- Franz Lemmermeyer, arXiv:math/0311306v1
- Three Fermat Trails to Elliptic Curves
- Ezra Brown, The College Mathematics Journal, Vol. 31, No. 3 (May, 2000), pp. 162-172
- Elliptic Curves
- John Stillwell, The American Mathematical Monthly, Vol. 102, No. 9 (Nov., 1995), pp. 831-837
- Taxicabs and Sums of Two Cubes
- Joseph H. SilvermanThe American Mathematical Monthly, Vol. 100, No. 4 (Apr., 1993), pp. 331-340
- Why Study Equations over Finite Fields?
- Neal Koblitz, Mathematics Magazine, Vol. 55, No. 3 (May, 1982), pp. 144-149
관련논문
- Heegner points and derivatives of L-series. II
- Gross, B.; Kohnen, W.; Zagier, D. (1987), Mathematische Annalen 278 (1–4): 497–562
- Gross, B.; Kohnen, W.; Zagier, D. (1987), Mathematische Annalen 278 (1–4): 497–562
- Heegner points and derivatives of L-series
- Gross, Benedict H.; Zagier, Don B. (1986), Inventiones Mathematicae 84 (2): 225–320
- Gross, Benedict H.; Zagier, Don B. (1986), Inventiones Mathematicae 84 (2): 225–320
- On the Conjecture of Birch and Swinnerton-Dyer for an Elliptic Curve of Rank 3
- Joe P. Buhler, Benedict H. Gross and Don B. Zagier, Mathematics of Computation, Vol. 44, No. 170 (Apr., 1985), pp. 473-481
- 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
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관련기사
- 네이버 뉴스 검색 (키워드 수정)