On Szpiro’s discriminant conjecture

Abstract: We consider generalizations of Szpiro’s classical discriminant conjecture to hyperelliptic curves over a number field \(K\) and to smooth, projective, and geometrically connected curves \(X\) over \(K\) of genus at least 1. The main results give effective exponential versions of the generalized conjectures for some curves, including all curves \(X\) of genus \(1\) or \(2\). In particular, we establish completely explicit exponential versions of Szpiro’s classical discriminant conjecture for elliptic curves over \(K\). The proofs use the theory of logarithmic forms and Arakelov theory for arithmetic surfaces.