A Point Counting Algorithm for Cyclic Covers of the Projective Line

TL;DR

This paper introduces a generalized Kedlaya-style point counting algorithm for cyclic covers of the projective line, extending previous methods to a broader class of curves with practical improvements and experimental validation.

Contribution

It generalizes the Gaudry-G"urel algorithm to cyclic covers where $y^r = f(x)$, with enhancements like exploiting automorphisms and refined $p$-adic bounds.

Findings

Algorithm effectively computes Weil polynomials for large genus cyclic covers.

Practical improvements lead to more efficient computations.

Experimental results demonstrate the algorithm's applicability to complex curves.

Abstract

We present a Kedlaya-style point counting algorithm for cyclic covers $y^r = f(x)$ over a finite field $\mathbb{F}_{p^n}$ with $p$ not dividing $r$, and $r$ and $\deg{f}$ not necessarily coprime. This algorithm generalizes the Gaudry-G\"urel algorithm for superelliptic curves to a more general class of curves, and has essentially the same complexity. Our practical improvements include a simplified algorithm exploiting the automorphism of $\mathcal{C}$, refined bounds on the $p$-adic precision, and an alternative pseudo-basis for the Monsky-Washnitzer cohomology which leads to an integral matrix when $p \geq 2r$. Each of these improvements can also be applied to the original Gaudry-G\"urel algorithm. We include some experimental results, applying our algorithm to compute Weil polynomials of some large genus cyclic covers.