"버치와 스위너톤-다이어 추측"의 두 판 사이의 차이

2015년 12월 25일 (금) 02:19 판

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

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

Coates-Wiles theorem

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

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.
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].
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.
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.
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

@@ 89번째 줄: / 89번째 줄: @@
 ==관련논문==
+* Bruin, Peter, and Andrea Ferraguti. “On $L$-Functions of Quadratic $\mathbb{Q}$-Curves.” arXiv:1511.09001 [math], November 29, 2015. http://arxiv.org/abs/1511.09001.
 * Ota, Kazuto. “Kato’s Euler System and the Mazur-Tate Refined Conjecture of BSD Type.” arXiv:1509.00682 [math], September 2, 2015. http://arxiv.org/abs/1509.00682.
 * Balakrishnan, Jennifer, Ishai Dan-Cohen, Minhyong Kim, and Stefan Wewers. ‘A Non-Abelian Conjecture of Birch and Swinnerton-Dyer Type for Hyperbolic Curves’. arXiv:1209.0640 [math], 4 September 2012. http://arxiv.org/abs/1209.0640.