"버치와 스위너톤-다이어 추측"의 두 판 사이의 차이

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<h5>역사</h5>
 
<h5>역사</h5>
  
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The Birch and Swinnerton-Dyer conjecture has been proved only in special cases :
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* The Birch and Swinnerton-Dyer conjecture has been proved only in special cases :<br>
  
 
# In 1976 [http://en.wikipedia.org/wiki/John_Coates_%28mathematician%29 John Coates] and [http://en.wikipedia.org/wiki/Andrew_Wiles Andrew Wiles] proved that if <em>E</em> is a curve over a number field <em>F</em> with complex multiplication by an imaginary quadratic field <em>K</em> of [http://en.wikipedia.org/wiki/Class_number_%28number_theory%29 class number] 1, <em>F=K</em> or '''Q''', and <em>L(E,1)</em> is not 0 then <em>E</em> has only a finite number of rational points. This was extended to the case where <em>F</em> is any finite abelian extension of <em>K</em> by Nicole Arthaud-Kuhman, who shared an office with Wiles when both were students of Coates at Stanford.
 
# In 1976 [http://en.wikipedia.org/wiki/John_Coates_%28mathematician%29 John Coates] and [http://en.wikipedia.org/wiki/Andrew_Wiles Andrew Wiles] proved that if <em>E</em> is a curve over a number field <em>F</em> with complex multiplication by an imaginary quadratic field <em>K</em> of [http://en.wikipedia.org/wiki/Class_number_%28number_theory%29 class number] 1, <em>F=K</em> or '''Q''', and <em>L(E,1)</em> is not 0 then <em>E</em> has only a finite number of rational points. This was extended to the case where <em>F</em> is any finite abelian extension of <em>K</em> by Nicole Arthaud-Kuhman, who shared an office with Wiles when both were students of Coates at Stanford.
58번째 줄: 58번째 줄:
 
# In 1999 [http://en.wikipedia.org/wiki/Andrew_Wiles Andrew Wiles], [http://en.wikipedia.org/wiki/Christophe_Breuil Christophe Breuil], [http://en.wikipedia.org/wiki/Brian_Conrad Brian Conrad], [http://en.wikipedia.org/wiki/Fred_Diamond Fred Diamond] and [http://en.wikipedia.org/wiki/Richard_Taylor_%28mathematician%29 Richard Taylor] proved that all elliptic curves defined over the rational numbers are modular (the [http://en.wikipedia.org/wiki/Taniyama-Shimura_theorem Taniyama-Shimura theorem]), which extends results 2 and 3 to all elliptic curves over the rationals.
 
# In 1999 [http://en.wikipedia.org/wiki/Andrew_Wiles Andrew Wiles], [http://en.wikipedia.org/wiki/Christophe_Breuil Christophe Breuil], [http://en.wikipedia.org/wiki/Brian_Conrad Brian Conrad], [http://en.wikipedia.org/wiki/Fred_Diamond Fred Diamond] and [http://en.wikipedia.org/wiki/Richard_Taylor_%28mathematician%29 Richard Taylor] proved that all elliptic curves defined over the rational numbers are modular (the [http://en.wikipedia.org/wiki/Taniyama-Shimura_theorem Taniyama-Shimura theorem]), which extends results 2 and 3 to all elliptic curves over the rationals.
  
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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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*  Nothing has been proved for curves with rank greater than 1, although there is extensive numerical evidence for the truth of the conjecture.<br>
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
* [[수학사연표 (역사)|수학사연표]]
 
* [[수학사연표 (역사)|수학사연표]]

2012년 8월 15일 (수) 14:01 판

이 항목의 스프링노트 원문주소

 

 

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
    \(L(s,E)=\prod_pL_p(s,E)^{-1}\)
    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 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.

 

 

메모

 

 

관련된 항목들

 

 

수학용어번역

 

 

사전 형태의 자료

 

 

expository

 

 

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