Rational Points in Weighted Projective Spaces over Finite Fields

TL;DR

This paper provides explicit formulas and combinatorial methods for counting rational points on weighted projective spaces over finite fields, confirming the rationality of their zeta functions.

Contribution

It establishes equivalences of rational point notions, derives explicit counting formulas, and confirms zeta function rationality for weighted projective spaces over finite fields.

Findings

Explicit formulas for rational point counts on weighted projective spaces.

Confirmation of the rationality of the zeta function for these spaces.

Decomposition of point counts into smooth and singular loci.

Abstract

We establish the equivalence of three notions of $\mathbb{F}_q$-rational points on weighted projective spaces $\mathbb{P}_{\mathbf{w}}^n$ and derive explicit combinatorial formulas for their enumeration, leveraging Burnside's lemma and gcd (greatest common divisor) computations. We further derive formulas for point counts under weight normalization, providing closed expressions for the singular and smooth loci. Our results confirm the rationality of the zeta function $Z(\mathbb{P}_{\mathbf{w}}^n, t)$ via a finite product formula and reveal a canonical multiplicative decomposition aligning with the stratification into smooth and singular loci. These contributions advance the arithmetic theory of weighted projective spaces over finite fields, with computational examples illustrating the formulas for specific weights and fields.