Lifting $L$-polynomials of genus 2 curves

Let $C$ be a genus $2$ curve over $\mathbb{Q}$. Harvey and Sutherland's implementation of Harvey's average polynomial-time algorithm computes the $\bmod \ p$ reduction of the numerator of the zeta function of $C$ at all good primes $p\leq B$ in $O(B\log^{3+o(1)}B)$ time, which is $O(\log^{4+o(1)} p)$ time on average per prime. Alternatively, their algorithm can do this for a single good prime $p$ in $O(p^{1/2}\log^{1+o(1)}p)$ time. While Harvey's algorithm can also be used to compute the full zeta function, no practical implementation of this step currently exists. In this article, we present an $O(\log^{2+o(1)}p)$ Las Vegas algorithm that takes the $\bmod \ p$ output of Harvey and Sutherland's implementation and outputs the full zeta function. We then benchmark our results against the fastest algorithms currently available for computing the full zeta function of a genus~$2$ curve, finding substantial speedups in both the average polynomial-time and single prime settings.