Counting points on surfaces in polynomial time

TL;DR

This paper introduces a randomized polynomial-time algorithm for computing the local zeta function of smooth projective surfaces over the rationals at large primes, solving a longstanding conjecture.

Contribution

It provides the first polynomial-time algorithm for this problem, confirming a conjecture by Couveignes and Edixhoven.

Findings

Algorithm runs in polynomial time in log p

Successfully computes local zeta functions for surfaces

Resolves a major open problem in computational number theory

Abstract

We present a randomised algorithm to compute the local zeta function of a fixed smooth, projective surface over $\mathbb{Q}$, at any large prime $p$ of good reduction. The runtime of our algorithm is polynomial in $\log p$, resolving a conjecture of Couveignes and Edixhoven.