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

2000/04/07 by Qiu, DeRong, Zhang, Xianke
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.math/0004189

Abstract

Let p, q be twin prime numbers with q-p=2 . Consider the elliptic curves E=E_σ: y² = x (x+σp)(x+σq) . (σ=± 1). E=E_σis also denoted as E₊ or E_- when σ= +1or -1.Here the Mordell-Weil group and the rank of the elliptic curve E over the Gauss field K=Q(√ -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.

Related

• Almost prime values of the order of elliptic curves over finite fields
• Analytic problems for elliptic curves
• On the analytic rank of the twin prime elliptic curve y²=x(x-2)(x-p)
• On the Selmer groups and Mordell-Weil groups of elliptic curves y² = x (x ± p) (x ± q) over imaginary quadratic number fields of class number one
• On the Rank of the Elliptic Curve y²=x(x-p)(x-2)