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.

References

Hindry and Silverman showed, using the theory of local heights, that Lang's conjecture follows from Szpiro's conjecture, i.e.\ the uniform boundedness of the Szpiro ratio .

— 100% of odd hyperelliptic Jacobians have no rational points of small height (2405.10224 - Laga et al., 16 May 2024) in Introduction, Relation to existing results