버치와 스위너톤-다이어 추측

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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.
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