# 타원곡선

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## 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(B. Mazur) 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

• 타원 곡선 E의 conductor가 N일 때, 다음과 같이 정의됨

$L(s,E)=\prod_pL_p(s,E)^{-1}$ 여기서 $L_p(s,E)=\left\{\begin{array}{ll} (1-a_p p^{-s}+p^{1-2s}), & \mbox{if }p\nmid N \\ (1-a_pp^{-s}), & \mbox{if }p||N \\ 1, & \mbox{if }p^2|N \end{array}\right.$

• 여기서 $$a_p$$는 유한체위에서의 해의 개수와 관련된 정수로 $$a_p=p+1-\#E(\mathbb{F}_p)$$ (위의 하세-베유 정리)
• 타원곡선의 L-함수 항목 참조

## 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

• 1908 포앵카레 E(Q) 는 아벨군이다
• 1922 모델 E(Q)는 유한생성아벨군이다 (Weil generalized )
• 1978 Mazur torsion part of E(Q)
• 수학사 연표