Mordell-Weil Groups and Selmer Groups of Two Types of Elliptic Curves, by Qiu Derong and Zhang Xianke
Abstract: Consider elliptic curves
$\ E=E_\sigma : \; y^2 = x (x+\sigma p)
(x+\sigma q),\; $ where$\ \sigma =\pm 1,\; $
$p$ and $ q$ are prime numbers with $p+2=q$.
(1) The Selmer groups $\ S^{(2)}(E/{\mathbf{Q}}),\;
S^{(\varphi)}(E/{\mathbf{Q})}$, and
$\ S^{(\widehat{\varphi})}(E/{\mathbf{Q})}\ $
are explicitly determined, e.g.,
$\ S^{(2)}(E_{+1}/{\mathbf{Q}})= $
$({\mathbf{Z}}/2{\mathbf{Z}})^2;\ $
$ ({\mathbf{Z}}/2{\mathbf{Z}})^3;\ $ or
$ ({\mathbf{Z}}/2{\mathbf{Z}})^4\ $
when $p\equiv 5; \ 1 $ or $3;\ $ or
$ 7 \ ({\mathrm{mod}} \ 8)$ respectively.
(2) When $p\equiv 5 (\ 3 ,\ 5$
for $\sigma =-1) \ ({\mathrm{mod}} \ 8),\ $
it is proved that the Mordell- Weil group
$\ E({\mathbf{Q})} \cong $
$ {\mathbf{Z}}/2{\mathbf{Z}}
\oplus{\mathbf{Z}}/2{\mathbf{Z}}\ $ having rank $0,\ $ and
Shafarevich-Tate group {\CC ':} $(E/{\mathbf{Q}} )[2]=0.\ $
(3) In any case, the sum of
rank$E({\mathbf{Q})}$ and dimension of
{\CC ':} $(E/{\mathbf{Q}} )[2] $ is given, e.g., $0;\ 1;\ 2 $
when $p\equiv 5; \ 1 $ or $3;\ 7 \ ({\mathrm{mod}} \ 8)$
for $\sigma =1$.
(4) The Kodaira symbol, the torsion subgroup $E(K)_{tors}$
for any number field $K$ , etc. are also obtained.
Qiu Derong and Zhang Xianke <xianke@tsinghua.edu.cn>