Lower bounds for the modified Szpiro ratio

TL;DR

This paper establishes explicit lower bounds for the modified Szpiro ratio of elliptic curves over rationals with prescribed torsion subgroups, demonstrating these bounds are optimal.

Contribution

It provides the first explicit lower bounds for the modified Szpiro ratio for elliptic curves with each torsion subgroup allowed by Mazur's theorem, and proves these bounds are sharp.

Findings

Lower bounds for the modified Szpiro ratio for each torsion subgroup

Bounds are proven to be sharp

Results apply to all elliptic curves with specified torsion

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve. The modified Szpiro ratio of $E$ is the quantity $\sigma_{m}(E) =\log\max\left\{ \left\vert c_{4}^{3}\right\vert ,c_{6}^{2}\right\} /\log N_{E}$ where $c_{4}$ and $c_{6}$ are the invariants associated to a global minimal model of $E$, and $N_{E}$ denotes the conductor of $E$. In this article, we show that for each of the fifteen torsion subgroups $T$ allowed by Mazur's Torsion Theorem, there is a rational number $l_{T}$ such that if $T\hookrightarrow E(\mathbb{Q}) _{\text{tors}}$, then $\sigma_{m}(E) >l_{T}$. We also show that this bound is sharp.