Counting points on varieties over finite fields of small characteristic

TL;DR

This paper introduces a deterministic polynomial-time algorithm for computing the zeta function of varieties over finite fields of small characteristic, enabling efficient calculation of rational point group orders on Jacobians.

Contribution

The paper presents a novel deterministic algorithm that efficiently computes zeta functions of varieties over finite fields of small characteristic, improving upon previous methods.

Findings

Polynomial-time algorithm for zeta function computation

Efficient method for Jacobian rational point group order

Applicable to varieties of fixed dimension

Abstract

We present a deterministic polynomial time algorithm for computing the zeta function of an arbitrary variety of fixed dimension over a finite field of small characteristic. One consequence of this result is an efficient method for computing the order of the group of rational points on the Jacobian of a smooth geometrically connected projective curve over a finite field of small characteristic.