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

الأصل · EN

Consider elliptic curves E=Eσ: y² = x (x+σp) (x+σq), where σ=± 1, p and q are prime numbers with p+2=q. (1) The Selmer groups S⁽²⁾(E/Q), S⁽ϕ⁾(E/Q), and S(ϕ)(E/Q) are explicitly determined, e.g., S⁽²⁾(E₊₁/Q)= (Z/2Z)²; (Z/2Z)³; or (Z/2Z)⁴ when p≡ 5; 1 or 3; or 7 (mod 8) respectively. (2) When p≡ 5 (3, 5 for σ=-1) (mod 8), it is proved that the Mordell-Weil group E(Q) Z/2Z ⊕Z/2Z having rank 0, and Shafarevich-Tate group ': (E/Q)[2]=0. (3) In any case, the sum of rankE(Q) and dimension of ': (E/Q)[2] is given, e.g., 0; 1; 2 when p≡ 5; 1 or 3; 7 (mod 8) for σ=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.