On arithmetic progressions on Edwards curves
Volume 167 / 2015
Acta Arithmetica 167 (2015), 117-132
MSC: Primary 11G05, 11G30; Secondary 11B25, 11D45, 14G05.
DOI: 10.4064/aa167-2-2
Abstract
Let $m\in {\mathbb {Z}}_{>0}$ and $a,q\in {\mathbb {Q}}$. Denote by $\mathcal {AP}_{m}(a,q)$ the set of rational numbers $d$ such that $a,a+q,\dots ,a+(m-1)q$ form an arithmetic progression in the Edwards curve $E_d : x^2+y^2=1+dx^2 y^2$. We study the set $\mathcal {AP}_{m}(a,q)$ and we parametrize it by the rational points of an algebraic curve.
