Counting points on curves using a map to P^1, II

TL;DR

This paper presents a new algorithm for computing the zeta function of algebraic curves over finite fields, extending previous methods to a broader class of curves with known lifts to characteristic zero.

Contribution

It introduces a generalized algorithm for zeta function computation applicable to all curves with known good lifts to characteristic zero, including complexity analysis and implementation.

Findings

Algorithm successfully computes zeta functions for a wide class of curves.

Complexity bounds are established for the new method.

Implementation results demonstrate practical efficiency.

Abstract

We introduce a new algorithm to compute the zeta function of a curve over a finite field. This method extends previous work of ours to all curves for which a good lift to characteristic zero is known. We develop all the necessary bounds, analyse the complexity of the algorithm and provide a complete implementation.