Mordell-Weil groups and Selmer groups of two types of elliptic curves

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 $S^{(\hat{\varphi })}(E/\mathbf{Q})$ 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. This paper is a revised version of ANT-0229.