Derive the height conjecture from Szpiro’s conjecture for a family of elliptic curves

Determine how to deduce Frey’s height conjecture for a family of elliptic curves from Szpiro’s conjecture established for that family, including control of the Archimedean component of the Faltings height.

Background

The paper explains that the minimal discriminant represents the non-Archimedean part of the global Arakelov-theoretic Faltings height, while the Archimedean contribution requires separate control.

Consequently, even a sharp instance of Szpiro’s conjecture for a family does not presently yield the corresponding height-conjecture bound. The paper identifies this deduction as an unresolved issue and later treats elliptic curves with integral j-invariant by proving a direct power-saving estimate for the Faltings height.