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., 2024) in Introduction, Relation to existing results

Frey proposed a stronger version of Szpiro's conjecture: There is an absolute constant $c>0$ such that for all elliptic curves $E$ over $Q$ we have $h(E) \le c \log N_E$.

— Power-saving bounds for Thue--Mahler and Mordell equations  (2608.23559 - Pasten, 24 Aug 2026) in Section 1, subsection “The height conjecture”