Mordell-Weil groups and Selmer groups of twin-prime elliptic curves

<正> Let E = Ea : y2 = x(x + ap)(x + σp) be elliptic curves, where σ = ?, p and q are primenumbers with p + 2 = q. (i) Selmer groups 5(2)(E/Q), S(?)(E/Q), and S(?)(E/Q) are explicitly determined, e.g. S(2)(E+1/Q) = (Z/2Z)2, (Z/2Z)3, and (Z/2Z)4 when p = 5, 1 (or 3), and 7(mod 8), respectively, (ii) When p = 5 (3,5 for σ = -1) (mod 8), it is proved that the Mordell-Weil group E(Q) = Z/2Z (?) Z/2Z, rankE(Q) = 0, and Shafarevich-Tate group Ⅲ (E/Q)[2] = 0. (iii) In any case, the sum of rank E(Q) and dimension of Ⅲ (E/Q)[2] is given, e.g. 0, 1, 2 when p = 5,1 (or 3), 7 (mod 8) for σ = 1. (iv) The Kodaira symbol, the torsion subgroup E(K)tors for any number field K, etc. are also obtained.