In mathematics, the Hasse–Witt matrix H of a non-singular algebraic curve C over a finite field F is the matrix of the Frobenius mapping (p-th power mapping where F has q elements, q a power of the prime number p) with respect to a basis for the differentials of the first kind.
This definition, as given in the introduction, is natural in classical terms, and is due to Helmut Hasse and Ernst Witt (1936).
The connection with the Hasse–Witt definition is by means of Serre duality, which for a curve relates that group to where ΩC = Ω1C is the sheaf of Kähler differentials on C. The p-rank of an abelian variety A over a field K of characteristic p is the integer k for which the kernel A[p] of multiplication by p has pk points.
The reason that the p-rank is lower is that multiplication by p on A is an inseparable isogeny: the differential is p which is 0 in K. By looking at the kernel as a group scheme one can get the more complete structure (reference David Mumford Abelian Varieties pp.
It is there a question of classifying the possible Artin–Schreier extensions of the function field F(C) (the analogue in this case of Kummer theory).