Elliptic curves with torsion group Z/8Z or Z/2Z x Z/6Z

TL;DR

This paper demonstrates the existence of infinite families of elliptic curves over Q with specific torsion groups, achieving higher ranks than previously known, by constructing parameterized families with positive rank.

Contribution

It constructs new families of elliptic curves with torsion groups Z/8Z and Z/2Z x Z/6Z, with ranks at least 2 and 3, improving upon earlier results.

Findings

Existence of families with rank ≥ 2 for Z/8Z and Z/2Z x Z/6Z.

Infinite families parameterized by elliptic curves with positive rank.

Construction of curves with rank at least 3.

Abstract

We show the existence of families of elliptic curves over Q whose generic rank is at least 2 for the torsion groups Z/8Z and Z/2Z x Z/6Z. Also in both cases we prove the existence of infinitely many elliptic curves, which are parameterized by the points of an elliptic curve with positive rank, with such torsion group and rank at least 3. These results represent an improvement of previous results by Campbell, Kulesz, Lecacheux, Dujella and Rabarison where families with rank at least 1 were constructed in both cases.