Counting points on superelliptic curves in average polynomial time

Andrew V. Sutherland

Published 2020-04-21Version 1

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 $y^m=f(x)$ with $m\ge 2$ and $f\in \Z[x]$ any squarefree polynomial of degree $d\ge 3$, along with a positive integer $N$. It can compute $\#X(\Fp)$ for all $p\le N$ not dividing $m\lc(f)\disc(f)$ in time $O(md^3 N\log^3 N\log\log N)$. 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~$p$.