A bound on the genus of a curve with Cartier operator of small rank

Ekedahl showed that the genus of a curve in characteristic $$p>0$$ p > 0 with zero Cartier operator is bounded by $$p(p-1)/2$$ p ( p - 1 ) / 2 . We show the bound $$p+p(p-1)/2$$ p + p ( p - 1 ) / 2 in case the rank of the Cartier operator is 1, improving a result of Re.

Residue Free Differentials and the Cartier Operator for Algebraic Function Fields of one Variable

Let K be a field of characteristic p > 0 and let A be a separably generated algebraic function field of one variable with K as its exact constant field. Throughout this paper we shall use the following notations to classify differentials of A/K:D(A) : the K-module of all differentials,G(A) : the K-module of all differentials of the first kind,R(A) : the K-module of all residue free differentials in the sense of Chevalley [2, p. 48],E*(A) : the K-module of all pseudo-exact differentials in the sense of Lamprecht [7, p. 363], (compare the definition with our Lemma 8).