Szpiro's conjecture when the denominator of the $j$-invariant is small

TL;DR

This paper proves Szpiro's conjecture for a class of elliptic curves over the rationals where the denominator of the $j$-invariant is relatively small compared to its numerator, specifically logarithmic in size.

Contribution

It establishes Szpiro's conjecture for elliptic curves with $j$-invariants having small denominators, a new case not previously verified.

Findings

Szpiro's conjecture holds for elliptic curves with small denominator of $j$-invariant.

The proof applies to curves with denominator logarithmic in size relative to numerator.

Provides new evidence supporting Szpiro's conjecture in specific cases.

Abstract

We prove Szpiro's conjecture for elliptic curves over the rationals having $j$-invariant with denominator of logarithmic size with respect to its numerator.