Counting points on surfaces in polynomial time

TL;DR

This work presents a randomized algorithm that computes the local zeta function $Z(X/\mathbb{F}_{q},T)$ of a fixed smooth, projective surface $X$ over a finite field in time polynomial in $\log q$, by explicitly determining the action of Frobenius on the étale cohomology groups with torsion coefficients $\mathrm{H}^{i}(X,\mu_{\ell})$ for primes $\ell=O(\log q)$. The approach fibers $X$ via a Lefschetz pencil, analyzes vanishing cycles, and isolates the Edixhoven subspace to control the Galois action on $\mathrm{H}^{2}$, while Puiseux expansions and cospecialisation provide explicit access to the monodromy. Key ingredients include computation of $\ell$-division polynomials for the generic fibre, robust handling of nodal fibres through vanishing cycles, and Hensel lifting to transfer subspaces from positive characteristic to characteristic zero. The results resolve a conjecture of Couveignes–Edixhoven in this dimension and have potential implications for Langlands program data, Brauer–Manin obstructions in arithmetic geometry, and cryptographic applications involving higher-dimensional varieties, while outlining avenues toward higher-dimensional generalizations and possible quantum enhancements.

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.

Theorems & Definitions (101)

• Theorem 1.1
• Remark
• Remark
• Theorem 2.1: Arithmetic on Jacobians via divisors
• proof
• Theorem 2.2: Huang-Ierardi
• proof
• Theorem 2.3: Pila
• proof
• Remark