Counting points on superelliptic curves in average polynomial time

We describe the practical implementation of an average polynomial-time algorithm for counting points on superelliptic curves defined over Q that is substantially faster than previous approaches. Our algorithm takes as input a superelliptic curves yᵐ=f(x) with m≥ 2 and f∈ Z[x] any squarefree polynomial of degree d≥ 3, along with a positive integer N. It can compute #X( Fₚ) for all p≤ N not dividing mlc(f)disc(f) in time O(md³ Nlog³ Nloglog N). It achieves this by computing the trace of the Cartier--Manin matrix of reductions of X. We can also compute the Cartier--Manin matrix itself, which determines the p-rank of the Jacobian of X and the numerator of its zeta function modulo~p.