"버치와 스위너톤-다이어 추측"의 두 판 사이의 차이

수학노트
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* <math>E(\mathbb{Q})=\mathbb{Z}^r \oplus E(\mathbb{Q})_{\operatorname{Tor}}</math>의 rank r은 <math>\operatorname{Ord}_{s=1}L(s,E)</math>와 같다<br>
 
* <math>E(\mathbb{Q})=\mathbb{Z}^r \oplus E(\mathbb{Q})_{\operatorname{Tor}}</math>의 rank r은 <math>\operatorname{Ord}_{s=1}L(s,E)</math>와 같다<br>
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Coates-Wiles theorem
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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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# In 1976 [http://en.wikipedia.org/wiki/John_Coates_(mathematician) 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_(number_theory) 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.
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# In 1983 [http://en.wikipedia.org/wiki/Benedict_Gross Benedict Gross] and [http://en.wikipedia.org/wiki/Don_Zagier Don Zagier] showed that if a [http://en.wikipedia.org/wiki/Modular_elliptic_curve modular elliptic curve] has a first-order zero at <em>s</em> = 1 then it has a rational point of infinite order; see [http://en.wikipedia.org/wiki/Gross%E2%80%93Zagier_theorem Gross–Zagier theorem].
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# In 1990 [http://en.wikipedia.org/wiki/Victor_Kolyvagin Victor Kolyvagin] showed that a modular elliptic curve <em>E</em> for which <em>L(E,1)</em> is not zero has rank 0, and a modular elliptic curve <em>E</em> for which <em>L(E,1)</em> has a first-order zero at <em>s</em> = 1 has rank 1.
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# In 1991 [http://en.wikipedia.org/wiki/Karl_Rubin Karl Rubin] showed that for elliptic curves defined over an imaginary quadratic field <em>K</em> with complex multiplication by <em>K</em>, if the <em>L</em>-series of the elliptic curve was not zero at <em>s=1</em>, then the <em>p</em>-part of the Tate-Shafarevich group had the order predicted by the Birch and Swinnerton-Dyer conjecture, for all primes <em>p > 7</em>.
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# 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_(mathematician) 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.
  
 
* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
* http://www.google.com/search?hl=en&tbs=tl:1&q=
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]<br>
 
* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]<br>
 
** http://www.research.att.com/~njas/sequences/?q=
 
** http://www.research.att.com/~njas/sequences/?q=
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<h5>expository</h5>
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* Wiles, A. "[http://www.claymath.org/millennium/Birch_and_Swinnerton-Dyer_Conjecture/BSD.pdf The Birch and Swinnerton-Dyer Conjecture]
  
 
 
 
 

2010년 3월 1일 (월) 09:03 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

 

 

 

재미있는 사실

 

 

 

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.

 

 

메모

 

 

관련된 항목들

 

 

수학용어번역

 

 

사전 형태의 자료

 

 

expository

 

 

관련논문

 

 

관련도서

 

 

관련기사

 

 

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