Lifting $L$-polynomials of genus 2 curves

Abstract: Let $C$ be a genus $2$ curve over $\mathbb{Q}$. For each odd prime $p$ of good reduction, we denote the numerator of the zeta function of $C$ at $p$ by $L_p(T)$.

Harvey and Sutherland's implementation of Harvey's average polynomial-time algorithm computes $L_p(T) \bmod \ p$ for 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 talk, I will present an $O(\log^{2+o(1)}p)$ Las Vegas algorithm that takes the $\bmod \ p$ output of Harvey and Sutherland's implementation and computes the full zeta function. I will also show benchmark results demonstrating substantial speedups compared to the fastest algorithms currently available for computing the full zeta function of a genus $2$ curve.

algebraic geometrynumber theory

Audience: researchers in the topic

( paper )

Boston University Number Theory Seminar