Cartier Points on Curves

Throughout this paper Y will denote a complete, nonsingular, and irreducible algebraic curve of genus g > 0 over the algebraically closed field k of characteristic p. Eventually we will assume that g ≥ 2. Denote by W the g-dimensional k-vector space of regular differentials of Y/k, which is just the vector space H(Y,ΩY/k). Also let K(Y ) denote the function field of Y , and let ΩK(Y )/k be the K(Y )-vector space of meromorphic differentials on Y . Recall (see, for example, [17, A2]) that there is a map C from ΩK(Y )/k to itself called the Cartier operator. The existence of this map only uses the fact that k is perfect. It preserves the space W of regular differentials. The Cartier operator is additive and also satisfies the identity