Discriminants and small points of cyclic covers

Abstract: Let K be a number field. We first consider a generalization of Szpiro’s discriminant conjecture to arbitrary smooth, projective and geometrically connected curves X/K of positive genus. Then we present an unconditional exponential version of this conjecture for cyclic covers of the projective line, and we discuss a related work (jointly with A. Javanpeykar) in which we established Szpiro’s small points conjecture for cyclic covers. We also plan to explain the proofs. They combine the theory of logarithmic forms with Arakelov theory for arithmetic surfaces.