"타원곡선"의 두 판 사이의 차이

2009년 10월 13일 (화) 08:17 판

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

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

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

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 (Z/2)*(Z/2n) for n=1,2,3,4

\(y^2=x^3-x\)

[/pages/2061314/attachments/2299029 MSP1975197gdf732cih44i50000361d01gd578fhc4a.gif]

\(y^2=4x^3-4x\)

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

@@ 22번째 줄: / 22번째 줄: @@
 * chord-tangent method
-* Mordell-Weil theorem
+*  유리수해에 대한 Mordell-Weil theorem<br>  <br>
@@ 30번째 줄: / 30번째 줄: @@
 <h5>rank와 torsion</h5>
+* 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 (Z/2)*(Z/2n) for n=1,2,3,4
@@ 113번째 줄: / 113번째 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
+*  Rational isogenies of prime degree<br>
+** Barry Mazur, Inventiones Math. 44 (1978), 129--162
 * http://www.jstor.org/action/doBasicSearch?Query=elliptic+curves