Computing representation matrices for the Frobenius on cohomology groups

Abstract In algebraic geometry, the Frobenius map F ⁎ on cohomology groups play an important role in the classification of algebraic varieties over a field of positive characteristic. In particular, representation matrices for F ⁎ give rise to many important invariants such as p-rank and a-number. Several methods for computing representation matrices for F ⁎ have been proposed for specific curves. In this paper, we present an algorithm to compute representation matrices for F ⁎ of general projective schemes over a perfect field of positive characteristic. We also propose an efficient algorithm specific to complete intersections; it requires to compute only certain coefficients in a power of a multivariate polynomial. Our algorithms shall derive fruitful applications such as computing Hasse-Witt matrices, and enumerating superspecial curves. In particular, the second algorithm provides a useful tool to judge the superspeciality of an algebraic curve, which is a key ingredient to prove main results in Kudo and Harashita (2017a) ; Kudo and Harashita , Kudo and Harashita, 2017b on the enumeration of superspecial genus-4 curves.