Counting points on superelliptic curves in average polynomial time

Abstract

We describe the practical implementation of an average polynomial-time algorithm for counting points on superelliptic curves defined over $\mathbb{Q}$ that is substantially faster than previous approaches. Our algorithm takes as input a superelliptic curves $y^m=f(x)$ with $m\ge 2$ and $f\in \mathbb{Z}[x]$ any squarefree polynomial of degree $d\ge 3$, along with a positive integer $N$. It can compute $\#X({\mathbb{F}}_p)$ for all $p\le N$ not dividing $m\mathrm{lc}(f)\mathrm{disc}(f)$ in time $O(m{d}^{3}N{\log }^{3}N\log \log N)$. 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~$p$.

minor corrections, 14 pages