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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 classi cation of algebraic varieties over a eld 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 com- puting representation matrices for F have been proposed for speci c curves. In this paper, we present an algorithm to compute representation matrices for F of general projective schemes over a perfect eld of positive character- istic. We also propose an efficient algorithm speci c 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 comput- ing 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,b, 2020) on the enumeration of superspecial genus-4 curves.
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https://hal.inria.fr/hal-02912348
Contributor : Alain Monteil <>
Submitted on : Wednesday, August 5, 2020 - 4:48:14 PM
Last modification on : Monday, May 17, 2021 - 12:00:05 PM
Long-term archiving on: : Monday, November 30, 2020 - 3:04:08 PM

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Momonari Kudo. Computing representation matrices for the Frobenius on cohomology groups. MEGA 2019 - International Conference on Effective Methods in Algebraic Geometry, Jun 2019, Madrid, Spain. ⟨hal-02912348⟩

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