Jeffrey D. Achter and Everett W. Howe: Hasse–Witt and Cartier–Manin matrices: A warning and a request.

Preprint version: arXiv:1710.10726 [math.NT].

Let X be a curve in positive characteristic. The Hasse–Witt matrix represents the action of the Frobenius operator on the cohomology group H1(X,OX). The Cartier–Manin matrix represents the action of the Cartier operator on the space of holomorphic differentials of X. The operators that these matrices represent are dual to one another, so the Hasse–Witt matrix and the Cartier–Manin matrix are related to one another, but they should not be viewed as being identical. There seems to be a fair amount of confusion in the literature about terminology, about whether matrices act on the left or the right, and about the proper formulæ for iterating semi-linear operators. Unfortunately, this confusion has led to the publication of incorrect results. In this paper we present the issues involved as clearly as we can, and we look through the literature to see where there may be problems. We encourage future authors to clearly distinguish between the Hasse–Witt and Cartier–Manin matrices, in the hope that further errors can be avoided.