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.