Class groups and L-series of function fields

Abstract

We consider a Galois covering Y X of complete nonsingular curves defined over a field of characteristic p > 0 and define generalized Hasse-Witt invariants. With the help of the representation of the endomorphisms of the jacobian by Witt vectors we compute these invariants to be the degree of L-series modulo p. We generalize results of Hasse and Witt, Katsurada, Manin, and Stichtenoth. The p-adic values of the L-series at t = 1 are computed.