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Counting points on smooth plane quartics

Edgar Costa, David Harvey, Andrew V. Sutherland

TL;DR

This work develops practical, scalable methods to count points on smooth plane quartics by computing Frobenius traces through Cartier–Manin matrices. It introduces three practical average-polynomial-time algorithms that reduce the Cartier–Manin computation to small 3×3 or compressed 16×16/28×28 matrix recurrences, enabling $O(p\log p\log\log p)$ or $O(p^{1/2}\log^2 p)$ time with modest space, and an $O(N\log^3 N)$-time, $O(N)$-space average algorithm across good primes $p\le N$. The core idea relies on a robust algebraic framework that translates the $F^m$ coefficient problem into low-dimensional linear recurrences via the differential equations $\partial_i(FG)=(m+1)(\partial_i F)G$ and carefully designed shift maps, yielding efficient computation of the Cartier–Manin matrix $A_p$ for plane quartics and, hence, $\#X(\mathbb{F}_p)=p+1-a_p$ with $a_p\equiv\operatorname{tr}(A_p)\pmod p$. The practical impact is substantial: these methods substantially extend the feasible range for point counting on genus-3 curves not cyclic covers and provide new pathways to compute $L_p(T)$ modulo $p$ and related arithmetic invariants.

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.

Counting points on smooth plane quartics

TL;DR

This work develops practical, scalable methods to count points on smooth plane quartics by computing Frobenius traces through Cartier–Manin matrices. It introduces three practical average-polynomial-time algorithms that reduce the Cartier–Manin computation to small 3×3 or compressed 16×16/28×28 matrix recurrences, enabling or time with modest space, and an -time, -space average algorithm across good primes . The core idea relies on a robust algebraic framework that translates the coefficient problem into low-dimensional linear recurrences via the differential equations and carefully designed shift maps, yielding efficient computation of the Cartier–Manin matrix for plane quartics and, hence, with . The practical impact is substantial: these methods substantially extend the feasible range for point counting on genus-3 curves not cyclic covers and provide new pathways to compute modulo and related arithmetic invariants.

Abstract

We present efficient algorithms for counting points on a smooth plane quartic curve modulo a prime . We address both the case where is defined over and the case where is defined over and is a prime of good reduction. We consider two approaches for computing , one which runs in time using space and one which runs in time using space. Both approaches yield algorithms that are faster in practice than existing methods. We also present average polynomial-time algorithms for that compute for good primes in time using space. These are the first practical implementations of average polynomial-time algorithms for curves that are not cyclic covers of , which in combination with previous results addresses all curves of genus . Our algorithms also compute Cartier-Manin/Hasse-Witt matrices that may be of independent interest.
Paper Structure (5 sections, 13 theorems, 69 equations, 1 figure, 4 tables)

This paper contains 5 sections, 13 theorems, 69 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. The Cartier matrix of a smooth plane curve
  3. Setup
  4. Computing Cartier--Manin matrices of a smooth plane quartic
  5. Performance comparisons

Key Result

Lemma 3.4

Let $h_0, \ldots, h_n$ be homogeneous polynomials in $K[x]$, of positive degree with no common zeros in $\mathbb{P}^n_K$. For $\ell \geqslant 0$, let Then, in $\mathbb{Z}[t]$ we have the identity

Figures (1)

  • Figure 1: Illustration for $p=7$. The target points $v$ in the interior are shown in black with $v^{(1)}$ at the top center, $v^{(2)}$ at the lower left, and $v^{(3)}$ at the lower right. The intermediate points $w$ are in blue, and the paths used to reach the target points $v$ are shown in gray.

Theorems & Definitions (46)

  • Definition 2.1
  • Remark 2.8
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Corollary 3.6
  • proof
  • Definition 3.11
  • ...and 36 more