Complexity of counting points on curves and the factor $P_1(T)$ of the zeta function of surfaces
Diptajit Roy, Nitin Saxena, Madhavan Venkatesh
TL;DR
This work analyzes the computational complexity of counting points on curves and higher-dimensional varieties over finite fields, introducing an Arthur–Merlin protocol to certify the zeta data for curves and extending it to certify the $P_{1}(T)$-factor for surfaces. By combining techniques from algebraic geometry (vanishing cycles, Lefschetz pencils, monodromy, Katz equidistribution) with complexity-theoretic frameworks (AM∩coAM, quantum algorithms), it reduces higher-dimensional problems to curve zeta computations and provides explicit complexity bounds, including a quantum poly$(D\log q)$-time algorithm for $P_{1}$ and a classical poly$(\log q)$-time algorithm for fixed degree. The results indicate that computing $P_{1}(T)$ is unlikely to be NP-hard and establish first efficient quantum procedures for these arithmetic invariants, with practical verification via randomized certificates in the AM setting. The work thus bridges computational number theory and complexity, offering new verifiable protocols and raising questions about extensions to higher cohomology and deterministic NP∩coNP certificates.
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($D\log q$)-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.