Families of elliptic curves with rational torsion points of even order

TL;DR

This paper constructs explicit families of elliptic curves with rational torsion points of specific even orders, advancing understanding of their structure and classification.

Contribution

It provides explicit constructions of versal families of elliptic curves with rational points of orders 4, 6, 8, 10, and 12, building on prior work.

Findings

Explicit families for each torsion order are constructed.

The families are versal, covering all cases up to isomorphism.

Enhances classification of elliptic curves with rational torsion points.

Abstract

This paper is a follow up of arXiv:1702.02255 [math.NT]. We construct explicitly versal families of elliptic curves with rational points of order 4, 6, 8, 10, 12 respectively.