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

LetE=E σ :y 2 =x(x+σp)(x+σq) be elliptic curves, where σ= ±1,p andq are prime numbers withp + 2= q. (i) Selmer groups S2(E/Q), S03D5(E/Q), and $S^{\widehat{(\varphi )}} \left( {E/Q} \right)$ are explicitly determined, e.g. S2(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) Whenp ≡ 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 rankE(Q) and dimension of III (E/Q)[2] is given, e.g. 0, 1, 2 whenp ≡ 5, 1 (or 3), 7 (mod 8) for σ= 1. (iv) The Kodaira symbol, the torsion subgroup E(K)tors for any number fieldK, etc. are also obtained.

Let E = Eσ : y2 = x(x + σp)(x + σq) be elliptic curves, where σ = ±1, p and q are primenumbers with p+2 = q. (i) 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, 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,symbol, the torsion subgroup E(K)tors for any number field K, etc. are also obtained.