Computational Algebra Seminar: Kyng -- Computing the zeta function of a curve using Harvey’s trace formula

Speaker: Madeleine Kyng (Sydney)
Title: Computing the zeta function of a curve using Harvey’s trace formula
Time and Place: 15:00-16:00, Thursday 14 November, SMRI Seminar Room
Abstract:
In this talk, I will describe a p-adic algorithm for computing the
zeta function that does not rely on any cohomology theory. This algorithm takes
as input a geometrically irreducible plane curve over a finite field and
outputs the zeta function of the nonsingular completion of that curve. Unlike
any other efficient algorithms for this problem, there are no smoothness or
nondegeneracy conditions imposed on the input plane curve. The algorithm is
based on Harvey’s algorithm for counting points on hypersurfaces, which is
applicable to a completely general hypersurface.

A version of this algorithm with time complexity O(p^2) is currently available
in MAGMA. A version with time complexity O(p) will soon be implemented.

I will compare the performance of this new algorithm with that of Tuitman’s
algorithm, and provide examples of inputs that cannot be handled by Tuitman’s
algorithm but can be handled by the new algorithm.