### 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>