Counting points on Cab curves using Monsky–Washnitzer cohomology

We describe an algorithm to compute the zeta function of any Cab curve over any finite field . The algorithm computes a p-adic approximation of the characteristic polynomial of Frobenius by computing in the Monsky–Washnitzer cohomology of the curve and thus generalizes Kedlayaʹs algorithm for hyperelliptic curves. For fixed p the asymptotic running time for a Cab curve of genus g over is O(g5+ n3+ ) and the space complexity is O(g3n3).