On the rank of elliptic curves with long arithmetic progressions
Volume 148 / 2017
Colloquium Mathematicum 148 (2017), 47-68
MSC: Primary 11G05; Secondary 11B25.
DOI: 10.4064/cm7036-9-2016
Published online: 17 February 2017
Abstract
We study the rank of elliptic curves associated to known curves of high arithmetic progressions. A set of rational points $(x_i, y_i)$ on an elliptic curve $E$ is said to be in arithmetic progression if the $x$-coordinates $x_i$ form an arithmetic progression. One of the motivations for finding curves with long progressions is to construct elliptic curves with high rank. We examine several curve families with long progressions and find their generic rank over $\mathbb {Q}(t)$, in addition to computing the rank of the specific curves with the longest progressions. We show that one of the infinite curve families with an arithmetic progression of length 12 has rank at least 8 over $\mathbb {Q}(t)$, and give generators.
