a-numbers of curves in Artin–Schreiercovers

Abstract

Let pi : Y -> X be a branched Z/pZ-cover of smooth, projective, geometrically connected curves over a perfect field of characteristic p > 0. We investigate the relationship between the a-numbers of Y and X and the ramification of the map pi. This is analogous to the relationship between the genus (respectively p-rank) of Y and X given the Riemann-Hurwitz (respectively Deuring-Shafarevich) formula. Except in special situations, the a-number of Y is not determined by the a-number of X and the ramification of the cover, so we instead give bounds on the a-number of Y. We provide examples showing our bounds are sharp. The bounds come from a detailed analysis of the kernel of the Cartier operator.