A remark on Artin-Schreier curves whose Hasse-Witt maps are the zero maps

Abstract

Let X be a complete non-singular algebraic curve over an algebraically closed field k of positive characteristic p. Let F: Ox―>Qx be the Frobenius homomorphism F(a)=ap, and denote the induced /^-linearmap H\X, Ox)-* H\X, Ox) again by F, which is called the Hasse-Witt map. The dimension of the semi-simple subspace H\X, Ox)s of H\X, Ox) is denoted by a(X) and called the £-rank of a curve X, which is equal to the jb-rank of the Jacobian variety of X. Let 7t:X-*Y be a /^-cyclicovering of complete non-singular curves over k. Then the Deuring-Safarevic formula is the following: