Lang's Height Conjecture and Szpiro's Conjecture

TL;DR

This paper demonstrates that a weaker form of Szpiro's conjecture, called prime-depleted, is sufficient to prove Lang's conjecture, establishing a new link between these important conjectures in number theory.

Contribution

It introduces the prime-depleted version of Szpiro's conjecture and proves that it implies Lang's conjecture, broadening the understanding of their relationship.

Findings

Prime-depleted Szpiro's conjecture suffices for Lang's conjecture

Establishes a new connection between conjectures in number theory

Provides a weaker assumption for proving Lang's conjecture

Abstract

It is known that Szpiro's conjecture, or equivalently the ABC-conjecture, implies Lang's conjecture giving a uniform lower bound for the canonical height of nontorsion points on elliptic curves. In this note we show that a significantly weaker version of Szpiro's conjecture, which we call "prime-depleted," suffices to prove Lang's conjecture.