Computing Zeta Functions of Cyclic Covers in Large Characteristic

TL;DR

This paper presents an efficient algorithm for computing the zeta functions of cyclic covers over finite fields of large characteristic, demonstrating practical performance through SageMath implementation.

Contribution

It introduces a new algorithm based on Gonçalves's generalization and Harvey's work, optimized for large characteristic fields, with proven practical efficiency.

Findings

Algorithm runs in time p^{1/2 + o(1)}

Successful implementation in SageMath

Effective on various example cases

Abstract

We describe an algorithm to compute the zeta function of a cyclic cover of the projective line over a finite field of characteristic $p$ that runs in time $p^{1/2 + o(1)}$. We confirm its practicality and effectiveness by reporting on the performance of our SageMath implementation on a range of examples. The algorithm relies on Gon\c{c}alves's generalization of Kedlaya's algorithm for cyclic covers, and Harvey's work on Kedlaya's algorithm for large characteristic.