Elliptic curves of twin-primes over Gauss field and Diophantine Equations

Let $p, q$ be twin prime numbers with $q-p=2$ . Consider the elliptic curves E=E_\sigma: y^2 = x (x+\sigma p)(x+\sigma q) . (\sigma =\pm 1). E=E_\sigma is also denoted as E_+ or E_- when \sigma = +1or $-1.Here the Mordell-Weil group and the rank of the elliptic curve E over the Gauss field K=Q(\sqrt -1) (and over the rational field Q is determined in several cases; and results on solutions of related Diophantine equations and simultaneous Pellian equations will be given. The arithmetic constructs over Q of the elliptic curve E have been studied in the last paper1, the Selmer groups are determined, results on Mordell-Weil group, rank, Shafarevich-Tate group, and torsion subgroups are also obtained.