"타원곡선"의 두 판 사이의 차이

2009년 10월 18일 (일) 18:05 판

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

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

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

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

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

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

\(y^2=x^3-x\)
[/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
\(2\omega=4\int_0^1\frac{dx}{\sqrt{1-x^4}}=B(1/2,1/4)=\frac{\Gamma(\frac{1}{2})\Gamma(\frac{1}{4})}{\Gamma(\frac{3}{4})}=5.24\cdots\)

Birch and Swinnerton-Dyer conjecture

@@ 102번째 줄: / 102번째 줄: @@
 * [[사각 피라미드 퍼즐]]<br>
 * [[congruent number 문제]]<br>
+* [[가우스의 class number one 문제]]<br>
@@ 133번째 줄: / 134번째 줄: @@
 <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련논문</h5>
-*   <br>
 * [http://arxiv.org/abs/math/0311306 Conics - a Poor Man's Elliptic Curves]<br>
 ** Franz Lemmermeyer, arXiv:math/0311306v1
@@ 140번째 줄: / 140번째 줄: @@
 * [http://www.jstor.org/stable/2687483 Three Fermat Trails to Elliptic Curves]<br>
 ** Ezra Brown, <cite>The College Mathematics Journal</cite>, Vol. 31, No. 3 (May, 2000), pp. 162-172
+* [http://www.jstor.org/stable/2007967 On the Conjecture of Birch and Swinnerton-Dyer for an Elliptic Curve of Rank 3]<br>
-#<br> On the Conjecture of Birch and Swinnerton-Dyer for an Elliptic Curve of Rank 3<br> # Joe P. Buhler, Benedict H. Gross and Don B. Zagier<br> # Mathematics of Computation, Vol. 44, No. 170 (Apr., 1985), pp. 473-481
+** 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<br>
-** Barry Mazur, Inventiones Math. 44 (1978), 129--162
+** Barry Mazur, Inventiones Math. 44 (1978), 129-162
 * http://www.jstor.org/action/doBasicSearch?Query=elliptic+curves