A study of Hasse-Witt matrices by local methods (Q1098904)

Let K be an algebraic function field in one variable of genus \(g\) with a perfect constant field k of prime characteristic \(p.\) In a classic paper \textit{H. Hasse} and \textit{E. Witt} [Monatsh. Math. Phys. 43, 477-492 (1936; Zbl 0013.34102)] described the abelian unramified extension of K of degree \(p\) in terms of a matrix depending on a choice of g points in general position and local parameters at them. In this paper we modify this procedure to obtain a description in terms of a matrix analogous to the one of Hasse-Witt but depending on a choice of only one rational prime and a local parameter at it. Such a local description not only simplifies practical computations but it is also of theoretical interest: our matrix, like the original one of Hasse-Witt, represents the Cartier operator, and we get bounds for the ranks of this operator and its iterates in terms of Weierstrass gap sequences at primes of K. We obtain also a generalization, for higher genus, of a formula of Atkin and Swinnerton-Dyer for elliptic function fields.