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Cartier points on curves

While studying ramified torsion points on curves in [9], Robert Coleman was naturally led to the study of points P on a curve in characteristic p having the property that the Cartier operator preserves the hyperplane of regular differentials vanishing at P . We call such points Cartier points, and investigate some of their basic properties. For example, we show that Cartier points, like Weierstrass points, are characterized by a certain “gap property”. During our investigation of Cartier points we strengthen some results of Coleman [8] and Ekedahl [13]. In particular, we obtain a new proof of Ekedahl’s theorem bounding the genus of a superspecial curve in characteristic p. We also obtain bounds for the number of Cartier points a curve can have, which in light of Coleman’s work has applications to studying torsion points on curves. Finally, we determine the complete set of Cartier points on some specific curves.