Lifting $L$-polynomials of genus $3$ curves

TL;DR

This work presents a practical, Laz Vegas-style lifting procedure for genus $3$ curves that takes the mod $p$ $L$-polynomial data from Costa–Harvey–Sutherland and reconstructs the full $L_p(T)$ by exploiting group operations on $\mathrm{Jac}(C)(\mathbb{F}_p)$ and $\mathrm{Jac}(C)(\mathbb{F}_{p^2})$. The method combines a three-step process: (i) bounding and enumerating candidate triples $(a_1,a_2,a_3)$ via Hasse–Weil/KS bounds, (ii) eliminating candidates with a baby-step giant-step search on the Jacobian, and (iii) resolving rare leftovers with a Las Vegas subroutine that uses Sylow subgroup checks. The paper also provides efficient, general implementations of Jacobian arithmetic for smooth plane quartics, including a hybrid approach that leverages Ritzenthaler’s typical divisor technique when possible and falls back to a naive method otherwise. Experimental results show favorable practical performance relative to existing algorithms for computing $L$-polynomials and lifting modulo $p$, and the lifting approach extends to hyperelliptic genus $3$ curves as well. Overall, the work delivers a concrete, efficient pathway to assemble $L$-functions of genus $3$ curves from modular data, enabling broader computational exploration of arithmetic conjectures tied to these objects.

Abstract

Let $C$ be a smooth plane quartic curve over $\mathbb{Q}$. Costa, Harvey and Sutherland provide an algorithm with an implementation, improving Harvey's average polynomial-time algorithm, to compute the $\bmod \ p$ reduction of the numerator of the zeta function of $C$ at all $p\leq B$, where $p$ is an odd prime of good reduction, in $O(B\log^{3+o(1)} N)$ time, which is $O(\log^{4+o(1)}p)$ time on average per prime. Alternatively, their algorithm can do this for a single prime $p$ of good reduction in $O(p^{1/2}\log^2p)$ time. While this algorithm can be used to compute the full zeta function, no implementation of this step currently exists. In this article, we provide an algorithm and an implementation for the group operation on the Jacobian of $C$ over $\mathbb{F}_p$, where $p$ is an odd prime of good reduction. We provide a Las Vegas algorithm that takes the $\bmod \ p$ result of Costa, Harvey and Sutherland's algorithm and uses it to compute the full zeta function. The expected running time of the algorithm is bounded by $O(p^{1/2+o(1)})$, and under heuristic assumptions, we prove an $O(p^{1/4+o(1)})$ bound on its average running time (over all inputs). Our lifting algorithm can also be applied to hyperelliptic curves of genus 3.

Theorems & Definitions (23)

• Lemma 2.1
• proof
• Proposition 2.2
• Definition 3.1
• Remark 3.2
• Lemma 5.1
• proof
• Proposition 5.2
• proof
• Proposition 5.3