Birch and Swinnerton-Dyer 추측

이 항목의 스프링노트 원문주소

 

 

개요

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

 

 

유리수해

• 

 

 

타원곡선의 L-함수

• Hasse-Weil 제타함수라고도 함

• 타원 곡선 E의 conductor가 N일 때, 다음과 같이 정의됨

  

  여기서 

  

• 여기서 는 유한체위에서의 해의 개수와 관련된 정수

 

 

추측

• 의 rank r은 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 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 Gross–Zagier 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 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.

• http://www.google.com/search?hl=en&tbs=tl:1&q=
• 수학사연표
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메모

 

 

관련된 항목들

• 타니야마-시무라 추측(정리)

 

 

수학용어번역

• 단어사전 http://www.google.com/dictionary?langpair=en|ko&q=
• 발음사전 http://www.forvo.com/search/

• 대한수학회 수학 학술 용어집

  • http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
• 남·북한수학용어비교
• 대한수학회 수학용어한글화 게시판

 

 

사전 형태의 자료

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture
• http://en.wikipedia.org/wiki/
• http://www.wolframalpha.com/input/?i=
• NIST Digital Library of Mathematical Functions

• The On-Line Encyclopedia of Integer Sequences

  • http://www.research.att.com/~njas/sequences/?q=

 

 

expository

• 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 (1–4): 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

• http://www.jstor.org/action/doBasicSearch?Query=
• http://www.ams.org/mathscinet
• http://dx.doi.org/

 

 

관련도서

• 도서내검색

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  • http://book.daum.net/search/contentSearch.do?query=

• 도서검색

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  • http://book.daum.net/search/mainSearch.do?query=
  • http://book.daum.net/search/mainSearch.do?query=

 

 

관련기사

• 네이버 뉴스 검색 (키워드 수정)

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