타원곡선

이 항목의 스프링노트 원문주소

타원곡선

개요

예

congruent number 문제
방정식 \(y^2=x^3-n^2x\) 이 등장
사각 피라미드 퍼즐
\(y^2=\frac{x(x+1)(2x+1)}{6}\)

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)

타니야마-시무라 추측(정리)

Birch and Swinnerton-Dyer 추측

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

수학사연표

complex multiplication

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.

Conics - a Poor Man's Elliptic CurvesFranz 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
Why Study Equations over Finite Fields? Neal Koblitz, Mathematics Magazine, Vol. 55, No. 3 (May, 1982), pp. 144-149

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