Complexity of counting points on curves and the factor $P_1(T)$ of the zeta function of surfaces

2025-11-04

AI Summary

This work addresses the efficient computation and interactive certification of rational point counts on curves and smooth projective surfaces over finite fields, along with the corresponding first Betti number factor $P_1(T)$ in their zeta functions—a problem lacking subexponential algorithms and whose complexity remains unknown. We introduce the first efficient Arthur–Merlin protocol for certifying curve point counts, Jacobian group structures, and full zeta functions, then extend it to surfaces to explicitly compute $P_1(T)$. Our algorithm runs in $mathrm{poly}(log q)$ time for fixed-degree surfaces and in quantum $mathrm{poly}(D log q)$ time generally, where $D$ bounds the degree. Key techniques include Weil-bound- and Riemann–Roch-based hashing for sampling; Lefschetz pencils to construct families of hyperplane sections; explicit instantiation of Deligne’s gcd theorem via vanishing cycle formalism and Katz’s equidistribution theorem; and quantum polynomial-time computation of curve zeta functions over extension fields. This yields the first nontrivial complexity upper bound for the problem.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Applications of cryptography
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

This article concerns the computational complexity of a fundamental problem in number theory: counting points on curves and surfaces over finite fields. There is no subexponential-time algorithm known and it is unclear if it can be $mathrm{NP}$-hard. Given a curve, we present the first efficient Arthur-Merlin protocol to certify its point-count, its Jacobian group structure, and its Hasse-Weil zeta function. We extend this result to a smooth projective surface to certify the factor $P_{1}(T)$, corresponding to the first Betti number, of the zeta function; by using the counting oracle. We give the first algorithm to compute $P_{1}(T)$ that is poly($log q$)-time if the degree $D$ of the input surface is fixed; and in quantum poly($Dlog q$)-time in general. Our technique in the curve case, is to sample hash functions using the Weil and Riemann-Roch bounds, to certify the group order of its Jacobian. For higher dimension varieties, we first reduce to the case of a surface, which is fibred as a Lefschetz pencil of hyperplane sections over $mathbb{P}^{1}$. The formalism of vanishing cycles, and the inherent big monodromy, enable us to prove an effective version of Deligne's `theoreme du pgcd'using the hard-Lefschetz theorem and an equidistribution result due to Katz. These reduce our investigations to that of computing the zeta function of a curve, defined over a finite field extension $mathbb{F}_{Q}/mathbb{F}_{q}$ of poly-bounded degree. This explicitization of the theory yields the first nontrivial upper bounds on the computational complexity.

Problem

• Analyzing computational complexity of counting points on curves and surfaces over finite fields
• Developing efficient protocols to certify point-counts and zeta functions of curves
• Computing the P₁(T) factor of zeta functions for surfaces with polynomial-time algorithms

Innovation

• Efficient Arthur-Merlin protocol for certifying point-counts
• Poly-time algorithm for computing zeta function factor P1(T)
• Technique using Weil bounds and Lefschetz pencil reduction