Elliptic curves with torsion groups $\mathbb{Z}/8\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$

TL;DR

This paper constructs and analyzes families of elliptic curves over rational function fields with specific torsion groups and ranks, providing examples with high rank and exploring their distribution properties.

Contribution

It introduces explicit families of elliptic curves with torsion groups Z/8Z and Z/2Z x Z/6Z, including high-rank examples and distribution analysis related to recent conjectures.

Findings

Existence of elliptic curves over Q(u) with rank 2 and specified torsion groups.

Construction of infinite families of elliptic curves with rank at least 3.

Analysis of root number distribution in these families.

Abstract

In this paper, we present details of seven elliptic curves over $\mathbb{Q}(u)$ with rank $2$ and torsion group $\mathbb{Z}/ 8\mathbb{Z}$ and five curves over $\mathbb{Q}(u)$ with rank $2$ and torsion group $\mathbb{Z}/ 2\mathbb{Z} \times \mathbb{Z}/ 6\mathbb{Z}$. We also exhibit some particular examples of curves with high rank over $\mathbb{Q}$ by specialization of the parameter. We present several sets of infinitely many elliptic curves in both torsion groups and rank at least $3$ parametrized by elliptic curves having positive rank. In some of these sets we have performed calculations about the distribution of the root number. This has relation with recent heuristics concerning the rank bound for elliptic curves by Park, Poonen, Voight and Wood.