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 many 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.