"타원곡선"의 두 판 사이의 차이

수학노트
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<h5>재미있는 사실</h5>
 
<h5>재미있는 사실</h5>
  
 
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Raussen and Skau: In the introduction to your<br> delightful book Rational Points on Elliptic Curves<br> that you coauthored with your earlier Ph.D. student<br> Joseph Silverman, you say, citing Serge Lang, that it<br> is possible to write endlessly on elliptic curves. Can<br> you comment on why the theory of elliptic curves<br> is so rich and how it interacts and makes contact<br> with so many different branches of mathematics?<br> Tate: For one thing, they are very concrete<br> objects. An elliptic curve is described by a cubic<br> polynomial in two variables, so they are very easy<br> to experiment with. On the other hand, elliptic<br> curves illustrate very deep notions. They are the<br> first nontrivial examples of abelian varieties. An<br> elliptic curve is an abelian variety of dimension<br> one, so you can get into this more advanced subject<br> very easily by thinking about elliptic curves.<br> On the other hand, they are algebraic curves.<br> They are curves of genus one, the first example<br> of a curve which isn’t birationally equivalent to a<br> projective line. The analytic and algebraic relations<br> which occur in the theory of elliptic curves and<br> elliptic functions are beautiful and unbelievably<br> fascinating. The modularity theorem stating that<br> every elliptic curve over the rational field can be<br> found in the Jacobian variety of the curve which<br> parametrizes elliptic curves with level structure its<br> conductor is mind-boggling.
  
 
 
 
 
181번째 줄: 181번째 줄:
 
<h5 style="margin: 0px; line-height: 2em;">expository articles</h5>
 
<h5 style="margin: 0px; line-height: 2em;">expository articles</h5>
  
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* Carella, N. A. 2011. “Topic In Elliptic Curves Over Finite Fields: The Groups of Points.” <em>1103.4560</em> (March 22). http://arxiv.org/abs/1103.4560.<br>  <br>
  
 
* [http://arxiv.org/abs/math/0311306 Conics - a Poor Man's Elliptic Curves]Franz Lemmermeyer, arXiv:math/0311306v1<br>
 
* [http://arxiv.org/abs/math/0311306 Conics - a Poor Man's Elliptic Curves]Franz Lemmermeyer, arXiv:math/0311306v1<br>

2011년 9월 24일 (토) 05:51 판

이 항목의 스프링노트 원문주소

 

 

개요

 

 

 

격자와 타원곡선
  • 타원곡선 \(y^2=4x^3-g_2(\tau)x-g_3(\tau)\)
    \(g_2(\tau) = 60G_4=60\sum_{ (m,n) \neq (0,0)} \frac{1}{(m+n\tau )^{4}}\)
    \(g_3(\tau) = 140G_6=140\sum_{ (m,n) \neq (0,0)} \frac{1}{(m+n\tau )^{6}}\)

 

 

주기
  • 타원곡선 \(y^2=(x-e_1)(x-e_2)(x-e_3)\)의 주기는 다음과 같이 정의된다
    \(\omega_1=2\int_{\infty}^{e_1}\frac{dx}{\sqrt{(x-e_1)(x-e_2)(x-e_3)}}\)
    \(\omega_2=2\int_{e_1}^{e_2}\frac{dx}{\sqrt{(x-e_1)(x-e_2)(x-e_3)}}\)
  • 타원곡선의 주기

 

 

군의 구조
  • chord-tangent method
  • 유리수해에 대한 Mordell theorem
    • 유리수체 위에 정의된 타원의 유리수해는 유한생성아벨군의 구조를 가짐
    • \(E(\mathbb{Q})=\mathbb{Z}^r \oplus E(\mathbb{Q})_{\operatorname{Tor}}\)
    • 여기서 \(E(\mathbb{Q})_{\operatorname{Tor}}\)는 \(E(\mathbb{Q})\)의 원소 중에서 order가 유한이 되는 원소들로 이루어진 유한군

 

 

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
  • \(E(\mathbb{Q})_{\operatorname{Tor}}\)는 오직 다음 열다섯가지 경우만이 가능하다
    크기가 1,2,3,4,5,6,7,8,9,10,12 (11은 불가)인 순환군 또는 \(\frac{\mathbb Z}{2\mathbb Z}\oplus \frac{\mathbb Z}{n\mathbb Z}\) 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

 

 

Hasse-Weil 정리
  • \(|\#E(\mathbb{F}_p)-p-1|\leq 2\sqrt{p}\)

 

 

 

타원곡선의 L-함수
  • http://cgd.best.vwh.net/home/flt/flt06.htm#intro
  • 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
  • Birch and Swinnerton-Dyer 추측 항목 참조

 

 

타니야마-시무라 추측(정리)

 

 

Birch and Swinnerton-Dyer 추측

 

 

타원곡선의 예

 

재미있는 사실

Raussen and Skau: In the introduction to your
delightful book Rational Points on Elliptic Curves
that you coauthored with your earlier Ph.D. student
Joseph Silverman, you say, citing Serge Lang, that it
is possible to write endlessly on elliptic curves. Can
you comment on why the theory of elliptic curves
is so rich and how it interacts and makes contact
with so many different branches of mathematics?
Tate: For one thing, they are very concrete
objects. An elliptic curve is described by a cubic
polynomial in two variables, so they are very easy
to experiment with. On the other hand, elliptic
curves illustrate very deep notions. They are the
first nontrivial examples of abelian varieties. An
elliptic curve is an abelian variety of dimension
one, so you can get into this more advanced subject
very easily by thinking about elliptic curves.
On the other hand, they are algebraic curves.
They are curves of genus one, the first example
of a curve which isn’t birationally equivalent to a
projective line. The analytic and algebraic relations
which occur in the theory of elliptic curves and
elliptic functions are beautiful and unbelievably
fascinating. The modularity theorem stating that
every elliptic curve over the rational field can be
found in the Jacobian variety of the curve which
parametrizes elliptic curves with level structure its
conductor is mind-boggling.

 

역사

 

 

 

 

 

관련된 항목들

 

 

수학용어번역

 

 

사전 형태의 자료

 

 

expository articles
  • Carella, N. A. 2011. “Topic In Elliptic Curves Over Finite Fields: The Groups of Points.” 1103.4560 (March 22). http://arxiv.org/abs/1103.4560.
     

 

 

관련논문

 

 

관련도서 및 추천도서

 

 

관련기사

 

 

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