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

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…
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