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Abstract

A Cartier point on a curve X in characteristic p is a point P such that the hyperplane of regular differentials on X vanishing at P is stable under the Cartier operator. In this paper we study some basic properties of Cartier points and as an application strengthen (and give a new proof of) a theorem of Ekedahl which says that the genus of a superspecial curve in characteristic p is at most p(p-1)/2. (A superspecial curve is one where the Cartier operator acts as zero on the regular differentials of the curve.) We also prove that there are infinitely many Cartier points on X if and only if X is superspecial, and we give examples of how Cartier points can be explicitly computed in practice. This work was originally motivated by a theorem of Coleman relating Cartier points to ramified torsion points on curves.