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 ${\sigma }_m(E)=\log \max \left\{\left|c_4^3\right|,c_6^2\right\}/\log N_E$ 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.

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

References in corpus (3)

• Minimal models of rational elliptic curves with non-trivial torsion
• Local data of rational elliptic curves with non-trivial torsion
• Good elliptic curves with a specified torsion subgroup