타원곡선

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간단한 소개

• congruent number 문제
• 사각 피라미드 퍼즐

 

 

 

격자와 타원곡선

\(y^2=4x^3-g_2(\tau)x-g_3\)

\(g_2(\tau) = 60G_4=60\sum_{ (m,n) \neq (0,0)} \frac{1}{(m+n\tau )^{4}}\)

\(g_3(\tau) = 140G_6=140\sum_{ (m,n) \neq (0,0)} \frac{1}{(m+n\tau )^{6}}\)

• 아이젠슈타인 급수(Eisenstein series)
  

 

 

군의 구조

• chord-tangent method
• 유리수해에 대한 Mordell theorem
  
  • 유리수체 위에 정의된 타원의 유리수해는 유한생성아벨군의 구조를 가짐
  • \(\mathbb{Z}^r \oplus \mathbb{T}\)
     
    

 

덧셈공식

• \(y^2=x^3+ax^2+bx+c\)위의 점 \(P=(x,y)\)에 대하여,
  \(2P\)의 \(x\)좌표는\(\frac{x^4-2bx^2-8cx-4ac+b^2}{4y^2}\) 로 주어진다
  

 

 

 

rank와 torsion

• the only possible torsion groups for elliptic curves over Q are the cyclic groups of order 1,2,3,4,5,6,7,8,9,10,12 [sic -- 11 is not possible] and
  \(\frac{\mathbb Z}{2\mathbb Z}\oplus \frac{\mathbb Z}{n\mathbb Z}\) for n=1,2,3,4
  
• 예) \(E_n : y^2=x^3-n^2x\)의 torsion은 \(\{(\infty,\infty), (0,0),(n,0),(-n,0)\}\)임

 

 

예

• \(y^2=x^3-x\)
  [/pages/2061314/attachments/2299029 MSP1975197gdf732cih44i50000361d01gd578fhc4a.gif]
  
• 유리수해
  \(E(\mathbb Q)=\{(\infty,\infty), (0,0),(1,0),(-1,0)\} \simeq \frac{\mathbb Z}{2\mathbb Z}\oplus \frac{\mathbb Z}{2\mathbb Z}\)
  
• 주기
  \(2\omega=4\int_0^1\frac{dx}{\sqrt{1-x^4}}=B(1/2,1/4)=\frac{\Gamma(\frac{1}{2})\Gamma(\frac{1}{4})}{\Gamma(\frac{3}{4})}=\frac{\Gamma(1/4)^2}{\sqrt{2\pi}}=5.24\cdots\)
  \(2\int_0^1\frac{dx}{\sqrt{x-x^3}}=B(1/2,1/4)=\frac{\Gamma(\frac{1}{2})\Gamma(\frac{1}{4})}{\Gamma(\frac{3}{4})}=5.24\cdots\)
  
• 모듈라 군, j-invariant and the singular moduli 의 special values 부분과 비교

 

 

 

 

 

 

 

 

 

 

역사

• 수학사연표

 

 

관련된 다른 주제들

• 타원적분
  
• lemniscate 곡선의 길이와 타원적분
  
• 정수계수 이변수 이차형식(binary integral quadratic forms)
  
• j-invariant
  
• 아이젠슈타인 급수(Eisenstein series)
  
• 베타적분
  
• 사각 피라미드 퍼즐
  
• congruent number 문제
  
• 가우스의 class number one 문제
  
• L-함수, 제타함수와 디리클레 급수
  
• periods
  
• 무리수와 초월수
  

 

 

수학용어번역

• 대한수학회 수학 학술 용어집
  
  • http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
• 대한수학회 수학용어한글화 게시판

 

 

사전 형태의 자료

• http://ko.wikipedia.org/wiki/타원곡선
• [1]http://en.wikipedia.org/wiki/elliptic_curve
• http://en.wikipedia.org/wiki/Mordell-Weil_theorem
• http://en.wikipedia.org/wiki/Heegner_point
• http://en.wikipedia.org/wiki/
• http://www.wolframalpha.com/input/?i=y^2=x^3-x
• http://www.wolframalpha.com/input/?i=
• NIST Digital Library of Mathematical Functions

 

 

expository articles

 

• Conics - a Poor Man's Elliptic Curves
  
  • Franz Lemmermeyer, arXiv:math/0311306v1
• Three Fermat Trails to Elliptic Curves
  
  • Ezra Brown, The College Mathematics Journal, Vol. 31, No. 3 (May, 2000), pp. 162-172
• Elliptic Curves
  
  • John Stillwell, The American Mathematical Monthly, Vol. 102, No. 9 (Nov., 1995), pp. 831-837
• Taxicabs and Sums of Two Cubes
  
  • Joseph H. SilvermanThe American Mathematical Monthly, Vol. 100, No. 4 (Apr., 1993), pp. 331-340

 

 

관련논문

• 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
• Rational isogenies of prime degree
  
  • Barry Mazur, Inventiones Math. 44 (1978), 129-162
• http://www.jstor.org/action/doBasicSearch?Query=elliptic+curves
• http://www.jstor.org/action/doBasicSearch?Query=congruent+number+problem
• http://www.jstor.org/action/doBasicSearch?Query=

 

 

관련도서 및 추천도서

• Introduction to elliptic curves and modular forms‎
  
  • Neal Koblitz - 1993
• Rational points on elliptic curves‎
  
  • Joseph H. Silverman, John Torrence Tate - 1992
  • 학부생의 입문용으로 좋은 책
• The Arithmetic of Elliptic Curves
  
  • Silverman, Joseph H. (1986), Graduate Texts in Mathematics, 106, Springer-Verlag
    
• 도서내검색
  
  • http://books.google.com/books?q=타원곡선
  • http://book.daum.net/search/contentSearch.do?query=타원곡선
  • http://book.daum.net/search/contentSearch.do?query=
• 도서검색
  
  • http://books.google.com/books?q=elliptic+curves
  • http://books.google.com/books?q=
  • http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
  • http://book.daum.net/search/mainSearch.do?query=

 

 

관련기사

• 네이버 뉴스 검색 (키워드 수정)
  
  • http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=타원곡선
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