### Elliptic Curves of Twin-Primes Over Gauss Field and Diophantine Equations, by DeRong QIU and XianKe ZHANG

Abstract: 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.

DeRong QIU and XianKe ZHANG <xianke@tsinghua.edu.cn , xzhang@maht.tsinghua.edu.cn>