A preprint version is also available: 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 *H*^{1}(*X*,*O*_{X}).
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.