On the rank of elliptic curves with long arithmetic progressions

Dustin Moody ¹   ; Arman Shamsi Zargar ²

¹ Computer Security Division National Institute of Standards and Technology 100 Bureau Drive Gaithersburg, MD 20899-8930, U.S.A.
² Young Researchers and Elite Club Ardabil Branch Islamic Azad University Ardabil, Iran

Colloquium Mathematicum, Tome 148 (2017) no. 1, pp. 47-68

Résumé

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.

Détail

DOI : 10.4064/cm7036-9-2016

Keywords: study rank elliptic curves associated known curves high arithmetic progressions set rational points elliptic curve said arithmetic progression x coordinates form arithmetic progression motivations finding curves long progressions construct elliptic curves high rank examine several curve families long progressions their generic rank mathbb addition computing rank specific curves longest progressions infinite curve families arithmetic progression length has rank least mathbb generators