Szpiro’s conjecture (uniform boundedness of the Szpiro ratio)

Determine whether the Szpiro ratio associated to elliptic curves over Q is uniformly bounded above by an absolute constant, thereby implying Lang’s conjecture.

Background

The authors explain that Lang’s conjecture follows from Szpiro’s conjecture, which asserts a uniform bound on the Szpiro ratio of elliptic curves over Q.

This connection situates Szpiro’s conjecture as a stronger statement whose resolution would settle Lang’s conjecture.