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 classication 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 specic 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 specic 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.
Origin : Files produced by the author(s)
Loading...