Mordell-Weil Groups and Selmer Groups of Two Types of Elliptic Curves, by DeRong QIU and XianKe ZHANG

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/Q), S^{(\varphi)}(E/Q), and S^{(\widehat{\varphi})}(E/Q) are explicitly determined, e.g., S^{(2)}(E_{+1}/Q)= (Z/2Z)^2; (Z}}/2Z)^3; or (Z/2Z)^4 when $p\equiv 5; 1 or 3; or 7 (mod 8) respectively. (2) When p\equiv 5 (3 ,5 for \sigma =-1) (mod 8), it is proved that the Mordell- Weil group E(Q) \cong Z/2Z \oplus Z/2Z having rank 0, and Shafarevich-Tate group \247\272(E/Q)[2]=0. (3) In any case, the sum of rankE(Q)and dimension of \247\272(E/Q)[2] is given, e.g., 0; 1; 2 when p\equiv 5; 1 or 3; 7 (mod 8) for \sigma =1. (4) The Kodaira symbol, the torsion subgroup E(K)_{tors} for any number field K , etc. are also obtained.

DeRong QIU and XianKe ZHANG < ,>