A Point Counting Algorithm for Cyclic Covers of the Projective Line

We present a Kedlaya-style point counting algorithm for cyclic covers y = f(x) over a finite field Fpn with p not dividing r, and r and deg f not necessarily coprime. This algorithm generalizes the Gaudry–Gürel 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 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 ≥ 2r. Each of these improvements can also be applied to the original Gaudry–Gürel algorithm. We include some experimental results, applying our algorithm to compute Weil polynomials of some large genus cyclic covers.