버치와 스위너톤-다이어 추측

개요

• 타원곡선의 rank는 잘 알려져 있지 않다
• Birch and Swinnerton-Dyer 추측은 타원곡선의 rank에 대한 밀레니엄 추측의 하나이다

유리수해

• \(E(\mathbb{Q})=\mathbb{Z}^r \oplus E(\mathbb{Q})_{\operatorname{Tor}}\)

타원곡선의 L-함수

• Hasse-Weil 제타함수라고도 함
• 타원 곡선 E의 conductor가 N일 때, 다음과 같이 정의됨\[L(s,E)=\prod_p L_p (s,E)^{-1}\] 여기서 \[L_p(s,E)=\left\{\begin{array}{ll}(1-a_pp^{-s}+p^{1-2s}), & \mbox{if }p \nmid N \\ (1-a_pp^{-s}), & \mbox{if }p||N \\ 1, & \mbox{if }p^2|N \end{array}\right.\]
• 여기서 \(a_p\)는 유한체위에서의 해의 개수와 관련된 정수

추측

• \(E(\mathbb{Q})=\mathbb{Z}^r \oplus E(\mathbb{Q})_{\operatorname{Tor}}\)의 rank r은 \(\operatorname{Ord}_{s=1}L(s,E)\)와 같다

Coates-Wiles theorem

역사

• The Birch and Swinnerton-Dyer conjecture has been proved only in special cases :

1. In 1976 John Coates and Andrew Wiles proved that if E is a curve over a number field F with complex multiplication by an imaginary quadratic field K of %28 number_theory %29 class number 1, F=K or Q, and L(E,1) is not 0 then E has only a finite number of rational points. This was extended to the case where F is any finite abelian extension of K by Nicole Arthaud-Kuhman, who shared an office with Wiles when both were students of Coates at Stanford.
2. In 1983 Benedict Gross and Don Zagier showed that if a modular elliptic curve has a first-order zero at s = 1 then it has a rational point of infinite order; see E2 %80 %93 Zagier_theorem Gross\[DashZagier theorem].
3. In 1990 Victor Kolyvagin showed that a modular elliptic curve E for which L(E,1) is not zero has rank 0, and a modular elliptic curve E for which L(E,1) has a first-order zero at s = 1 has rank 1.
4. In 1991 Karl Rubin showed that for elliptic curves defined over an imaginary quadratic field K with complex multiplication by K, if the L-series of the elliptic curve was not zero at s=1, then the p-part of the Tate-Shafarevich group had the order predicted by the Birch and Swinnerton-Dyer conjecture, for all primes p > 7.
5. In 1999 Andrew Wiles, Christophe Breuil, Brian Conrad, Fred Diamond and %28 mathematician %29 Richard Taylor proved that all elliptic curves defined over the rational numbers are modular (the Taniyama-Shimura theorem), which extends results 2 and 3 to all elliptic curves over the rationals.

• Nothing has been proved for curves with rank greater than 1, although there is extensive numerical evidence for the truth of the conjecture.
• 수학사 연표

관련된 항목들

• 타니야마-시무라 추측(정리)

수학용어번역

• Birch - 발음사전 Forvo
• Swinnerton-Dyer - 발음사전 Forvo

사전 형태의 자료

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture

리뷰, 에세이, 강의노트

• Wiles, A. "The Birch and Swinnerton-Dyer Conjecture

관련논문

• Finiteness of E(Q) and Sha(E, Q) for a subclass of Weil curves
  • V. Kolyvagin, (transl.) Math. USSR-Izv. 32 (1989), no. 3, 523--541
• Heegner points and derivatives of L-series. II
  • Gross, B.; Kohnen, W.; Zagier, D. (1987), Mathematische Annalen 278 (1\[Dash]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