Good elliptic curves with a specified torsion subgroup

TL;DR

This paper proves the existence of infinitely many 'good' elliptic curves over rationals with each of the fifteen torsion subgroups allowed by Mazur's Theorem, extending previous results on full 2-torsion.

Contribution

It generalizes Masser's Theorem by constructing infinitely many good elliptic curves for all torsion subgroups permitted by Mazur's classification.

Findings

Existence of infinitely many good elliptic curves for each torsion subgroup

Constructive methods used for proof

Extension of Masser's Theorem to all torsion types

Abstract

An elliptic curve $E$ over $\mathbb{Q}$ is said to be good if $N_{E}^{6}<\max\!\left\{ \left\vert c_{4}^{3}\right\vert ,c_{6}^{2}\right\} $ where $N_{E}$ is the conductor of $E$ and $c_{4}$ and $c_{6}$ are the invariants associated to a global minimal model of $E$. In this article, we generalize Masser's Theorem on the existence of infinitely many good elliptic curves with full $2$-torsion. Specifically, we prove via constructive methods that for each of the fifteen torsion subgroups $T$ allowed by Mazur's Torsion Theorem, there are infinitely many good elliptic curves $E$ with $E\!\left(\mathbb{Q}\right) _{\text{tors}}\cong T$.