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

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.