Hasse-Witt and Cartier-Manin matrices: A warning and a request

TL;DR

This paper clarifies the relationship between Hasse-Witt and Cartier-Manin matrices for curves in positive characteristic, addressing widespread confusion and correcting misconceptions to improve future research accuracy.

Contribution

It provides a clear analysis of the differences and relationships between Hasse-Witt and Cartier-Manin matrices, highlighting common errors and offering guidance for future work.

Findings

Identifies sources of confusion in the literature

Clarifies the mathematical relationship between the matrices

Warns against common misconceptions and errors

Abstract

Let X be a curve in positive characteristic. A Hasse--Witt matrix for X is a matrix that represents the action of the Frobenius operator on the cohomology group H^1(X,O_X) with respect to some basis. A Cartier--Manin matrix for X is a matrix that represents the action of the Cartier operator on the space of holomorphic differentials of X with respect to some basis. The operators that these matrices represent are adjoint to one another, so Hasse--Witt matrices and the Cartier--Manin matrices are related to one another, but there seems to be a fair amount of confusion in the literature about the exact nature of this relationship. This confusion arises from differences in terminology, from differing conventions about whether matrices act on the left or on the right, and from misunderstandings about the proper formulae for iterating semilinear 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 Hasse--Witt and Cartier--Manin matrices, in the hope that further errors can be avoided.