방정식 이 등장
방정식 이 등장
타원곡선
타원곡선 의 주기는 다음과 같이 정의된다
유리수해에 대한 Mordell theorem
위의 점
에 대하여,
의
좌표는
로 주어진다
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 for n=1,2,3,4
타원 곡선 E의 conductor가 N일 때, 다음과 같이 정의됨
여기서
여기서 는 유한체위에서의 해의 개수와 관련된 정수로
(위의 Hasse-Weil 정리
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.
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
Why Study Equations over Finite Fields? Neal Koblitz, Mathematics Magazine, Vol. 55, No. 3 (May, 1982), pp. 144-149
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
Introduction to elliptic curves and modular forms
Rational points on elliptic curves
The Arithmetic of Elliptic Curves
Silverman, Joseph H. (1986), Graduate Texts in Mathematics, 106, Springer-Verlag
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