Almost all elliptic curves with prescribed torsion have Szpiro ratio close to the expected value

TL;DR

This paper shows that most elliptic curves with a given torsion subgroup have Szpiro ratios near the expected value, providing bounds and using polynomial bounding techniques to extend previous results.

Contribution

It extends the understanding of Szpiro ratios for elliptic curves with prescribed torsion, establishing bounds and probabilistic behavior for a broad class of curves.

Findings

Almost all elliptic curves with prescribed torsion have Szpiro ratio close to the expected value.

Bounds for Szpiro ratios are established for curves in certain families.

The polynomial bounding technique is adapted to analyze elliptic curve properties.

Abstract

We demonstrate that almost all elliptic curves over $\mathbb{Q}$ with prescribed torsion subgroup, when ordered by naive height, have Szpiro ratio arbitrarily close to the expected value. We also provide upper and lower bounds for the Szpiro ratio that hold for almost all elliptic curves in certain one-parameter families. The results are achieved by proving that, given any multivariate polynomial within a general class, the absolute value of the polynomial over an expanding box is typically bounded by a fixed power of its radical. The proof adapts work of Fouvry--Nair--Tenenbaum, which shows that almost all elliptic curves have Szpiro ratio close to $1$.