Lower bounds for the modified Szpiro ratio

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.Comment: 15 pages; incorporates referee's suggestions; sharpness of lower bounds is no longer conditional on the abc conjecture; final version to appear in Acta Arithmetic