Counting points on smooth plane quartics

TL;DR

This paper introduces new, faster algorithms for counting points on smooth plane quartic curves over finite fields, applicable both over finite fields and number fields, with practical implementations for genus 3 curves.

Contribution

The paper presents the first practical average polynomial-time algorithms for counting points on non-cyclic genus 3 curves, improving efficiency over existing methods.

Findings

Algorithms run in $O(p ext{log} p ext{log} ext{log} p)$ and $O(p^{1/2} ext{log}^2 p)$ time.

Practical implementations outperform previous methods.

First practical average polynomial-time algorithms for non-cyclic genus 3 curves.

Abstract

We present efficient algorithms for counting points on a smooth plane quartic curve $X$ modulo a prime $p$. We address both the case where $X$ is defined over $\mathbb F_p$ and the case where $X$ is defined over $\mathbb Q$ and $p$ is a prime of good reduction. We consider two approaches for computing $\#X(\mathbb F_p)$, one which runs in $O(p\log p\log\log p)$ time using $O(\log p)$ space and one which runs in $O(p^{1/2}\log^2\!p)$ time using $O(p^{1/2}\log p)$ space. Both approaches yield algorithms that are faster in practice than existing methods. We also present average polynomial-time algorithms for $X/\mathbb Q$ that compute $\#X(\mathbb F_p)$ for good primes $p\le N$ in $O(N\log^3\! N)$ time using $O(N)$ space. These are the first practical implementations of average polynomial-time algorithms for curves that are not cyclic covers of $\mathbb P^1$, which in combination with previous results addresses all curves of genus $g\le 3$. Our algorithms also compute Cartier-Manin/Hasse-Witt matrices that may be of independent interest.