타원곡선
개요
- a smooth, projective algebraic curve of genus one, on which there is a specified point O
- abelian variety
- 19세기 타원함수론과 함께 발전
- 현대 정수론의 중요한 연구주제
- 유리수체 위에 정의된 타원곡선에 대한 타니야마-시무라 추측(정리) 으로 페르마의 마지막 정리 가 해결
- 타원곡선에 대한 Birch and Swinnerton-Dyer 추측 은 클레이 연구소가 선정한 7개의 밀레니엄 문제 중 하나
예
- 합동수 문제 (congruent number problem) 방정식 \(y^2=x^3-n^2x\) 이 등장
- 사각 피라미드 퍼즐\[y^2=\frac{x(x+1)(2x+1)}{6}\]
격자와 타원곡선
- 타원곡선 \(y^2=4x^3-g_2(\tau)x-g_3(\tau)\)\[g_2(\tau) = 60G_4=60\sum_{ (m,n) \neq (0,0)} \frac{1}{(m+n\tau )^{4}}\]\[g_3(\tau) = 140G_6=140\sum_{ (m,n) \neq (0,0)} \frac{1}{(m+n\tau )^{6}}\]
주기
- 타원곡선 \(y^2=(x-e_1)(x-e_2)(x-e_3)\)의 주기는 다음과 같이 정의된다\[\omega_1=2\int_{\infty}^{e_1}\frac{dx}{\sqrt{(x-e_1)(x-e_2)(x-e_3)}}\]\[\omega_2=2\int_{e_1}^{e_2}\frac{dx}{\sqrt{(x-e_1)(x-e_2)(x-e_3)}}\]
- 타원곡선의 주기
군의 구조
- chord-tangent method
- 유리수해에 대한 Mordell theorem
- 유리수체 위에 정의된 타원의 유리수해는 유한생성아벨군의 구조를 가짐
- \(E(\mathbb{Q})=\mathbb{Z}^r \oplus E(\mathbb{Q})_{\operatorname{Tor}}\)
- 여기서 \(E(\mathbb{Q})_{\operatorname{Tor}}\)는 \(E(\mathbb{Q})\)의 원소 중에서 order가 유한이 되는 원소들로 이루어진 유한군
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
- \(E(\mathbb{Q})_{\operatorname{Tor}}\)는 오직 다음 열다섯가지 경우만이 가능하다(B. Mazur) 크기가 1,2,3,4,5,6,7,8,9,10,12 (11은 불가)인 순환군 또는 \(\frac{\mathbb Z}{2\mathbb Z}\oplus \frac{\mathbb Z}{n\mathbb Z}\) 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
Hasse-Weil 정리
- \(|\#E(\mathbb{F}_p)-p-1|\leq 2\sqrt{p}\)
타원곡선의 L-함수
- 타원 곡선 E의 conductor가 N일 때, 다음과 같이 정의됨
\[L(s,E)=\prod_pL_p(s,E)^{-1}\] 여기서 \[L_p(s,E)=\left\{\begin{array}{ll} (1-a_p p^{-s}+p^{1-2s}), & \mbox{if }p\nmid N \\ (1-a_pp^{-s}), & \mbox{if }p||N \\ 1, & \mbox{if }p^2|N \end{array}\right.\]
- 여기서 \(a_p\)는 유한체위에서의 해의 개수와 관련된 정수로 \(a_p=p+1-\#E(\mathbb{F}_p)\) (위의 하세-베유 정리)
- 타원곡선의 L-함수 항목 참조
타니야마-시무라 추측(정리)
Birch and Swinnerton-Dyer 추측
타원곡선의 예
재미있는 사실
Raussen and Skau: In the introduction to your delightful book Rational Points on Elliptic Curves that you coauthored with your earlier Ph.D. student Joseph Silverman, you say, citing Serge Lang, that it is possible to write endlessly on elliptic curves. Can you comment on why the theory of elliptic curves is so rich and how it interacts and makes contact with so many different branches of mathematics? Tate: For one thing, they are very concrete objects. An elliptic curve is described by a cubic polynomial in two variables, so they are very easy to experiment with. On the other hand, elliptic curves illustrate very deep notions. They are the first nontrivial examples of abelian varieties. An elliptic curve is an abelian variety of dimension one, so you can get into this more advanced subject very easily by thinking about elliptic curves.
On the other hand, they are algebraic curves. They are curves of genus one, the first example of a curve which isn’t birationally equivalent to a projective line. The analytic and algebraic relations which occur in the theory of elliptic curves and elliptic functions are beautiful and unbelievably fascinating. The modularity theorem stating that every elliptic curve over the rational field can be found in the Jacobian variety of the curve which parametrizes elliptic curves with level structure its conductor is mind-boggling.
역사
- 1908 포앵카레 E(Q) 는 아벨군이다
- 1922 모델 E(Q)는 유한생성아벨군이다 (Weil generalized )
- 1978 Mazur torsion part of E(Q)
- 수학사 연표
관련된 항목들
- complex multiplication
- 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
리뷰, 에세이, 강의노트
- Carella, N. A. 2011. “Topic In Elliptic Curves Over Finite Fields: The Groups of Points.” 1103.4560 (March 22). http://arxiv.org/abs/1103.4560.
- Conics - a Poor Man's Elliptic CurvesFranz 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
- Jennifer S. Balakrishnan, Wei Ho, Nathan Kaplan, Simon Spicer, William Stein, James Weigandt, Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks, arXiv:1602.01894 [math.NT], February 05 2016, http://arxiv.org/abs/1602.01894
- Berghoff, Christian J. “Elliptic Gau{\ss} Sums and Schoof’s Algorithm.” arXiv:1601.03227 [math], January 13, 2016. http://arxiv.org/abs/1601.03227.
- Müller, J. Steffen, and Michael Stoll. “Computing Canonical Heights on Elliptic Curves in Quasi-Linear Time.” arXiv:1509.08748 [math], September 29, 2015. http://arxiv.org/abs/1509.08748.
- Buckley, Anita. “Indecomposable Matrices Defining Plane Cubics.” arXiv:1510.00133 [math], October 1, 2015. http://arxiv.org/abs/1510.00133.
- Guitart, Xavier, Marc Masdeu, and Mehmet Haluk Sengun. “Uniformization of Modular Elliptic Curves via P-Adic Periods.” arXiv:1501.02936 [math], January 13, 2015. http://arxiv.org/abs/1501.02936.
- Alpoge, Levent. “The Average Number of Integral Points on Elliptic Curves Is Bounded.” arXiv:1412.1047 [math], December 2, 2014. http://arxiv.org/abs/1412.1047.
- Rubin, Karl, and Alice Silverberg. “Ranks of Elliptic Curves.” Bulletin of the American Mathematical Society 39, no. 4 (2002): 455–74. doi:10.1090/S0273-0979-02-00952-7.
- Heegner points and derivatives of L-series. II
- 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
- 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
관련도서
- 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
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