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^{(\phi)}(E/{\mathbf{Q})}$, and $\ S^{(\hat{\phi})}(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.