Computing zeta functions of algebraic curves using Harvey's trace formula

TL;DR

This paper introduces a novel algorithm leveraging Harvey's trace formula to efficiently compute the zeta function of algebraic curves over finite fields, improving upon existing methods like Tuitman's algorithm.

Contribution

The paper presents a new method that uses Harvey's trace formula for more efficient zeta function computation of algebraic curves, including correction techniques at singular points.

Findings

Implementation in MAGMA shows improved performance over Tuitman's algorithm.

The method accurately computes zeta functions for various algebraic curves.

Demonstrates practical applicability with concrete examples.

Abstract

We present a new method for computing the zeta function of an algebraic curve over a finite field. The algorithm relies on a trace formula of Harvey to count points on a plane model of the curve. The zeta function of the curve is then obtained by making corrections at singular points. We report on an implementation and provide some examples in MAGMA which demonstrate an improvement over Tuitman's algorithm.