Computing representation matrices for the action of Frobenius on cohomology groups. (March 2022)

• Record Type: Journal Article
• Authors: Kudo, Momonari
• Is Part Of: Journal of symbolic computation. Volume 109(2022)
• Page Start: 441
• Page End: 464
• Publication Date: 2022-03
• Subjects: Cohomology groups -- Frobenius maps -- Hasse-Witt matrices; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2020.07.015
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: In algebraic geometry, the Frobenius map (or called Frobenius action, or Frobenius operator ) F ⁎ on cohomology groups plays 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 for complete intersections 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, b, 2020) on the enumeration of superspecial genus-4 curves.