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Rational Points in Weighted Projective Spaces over Finite Fields

Sajad Salami, Tanush Shaska

TL;DR

This work develops a coherent framework for counting $ inance{F}_q$-rational points on weighted projective spaces $ inance{P}_{oldsymbol w}^n$ over finite fields, reconciling three natural definitions of rational points and deriving Burnside-based formulas via gcd stratification. It then analyzes the associated zeta function $Z( inance{P}_{oldsymbol w}^n,t)$, proving rationality and delivering a canonical decomposition aligned with the smooth and singular stratification, while showing the Weil RH component holds but a standard functional equation generally fails due to singularities and weight data. The normalization of weights is shown not to preserve zeta functions, highlighting the dependence on weighted embedding; the results extend to weighted hypersurfaces and illuminate arithmetic properties of singular orbifolds with potential coding-theoretic applications. Overall, the paper provides explicit, computable point-counting formulas, a stratified zeta-theoretic interpretation, and avenues for applying these techniques to broader classes of weighted varieties and quantum code constructions.

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.

Rational Points in Weighted Projective Spaces over Finite Fields

TL;DR

This work develops a coherent framework for counting -rational points on weighted projective spaces over finite fields, reconciling three natural definitions of rational points and deriving Burnside-based formulas via gcd stratification. It then analyzes the associated zeta function , proving rationality and delivering a canonical decomposition aligned with the smooth and singular stratification, while showing the Weil RH component holds but a standard functional equation generally fails due to singularities and weight data. The normalization of weights is shown not to preserve zeta functions, highlighting the dependence on weighted embedding; the results extend to weighted hypersurfaces and illuminate arithmetic properties of singular orbifolds with potential coding-theoretic applications. Overall, the paper provides explicit, computable point-counting formulas, a stratified zeta-theoretic interpretation, and avenues for applying these techniques to broader classes of weighted varieties and quantum code constructions.

Abstract

We establish the equivalence of three notions of -rational points on weighted projective spaces 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 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.

Paper Structure

This paper contains 18 sections, 24 theorems, 109 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Weighted Projective Spaces and Rational Points
  3. The notion of $\mathbb{F}_q$-Rational point on weighted spaces
  4. Number of $\mathbb{F}_q$-rational Points on $\mathbb{P}_\mathbf{w}^n$
  5. Weight Normalization and $\mathbb{F}_q$-Rational Points
  6. Number of Singular $\mathbb{F}_q$-Rational Points on $\mathbb{P}_\mathbf{w}^n$
  7. Counting Rational Points on Weighted Varieties
  8. The Zeta Function of Weighted Projective Space
  9. The Zeta Function Standard Projective Space
  10. Rationality of the Zeta Function for $\mathbb{P}_{\mathbf{w}}^{n}$
  11. Decomposition of the Zeta Function
  12. Weil Conjectures for $\mathbb{P}_{\mathbf{w}}^{n}$
  13. Decomposition over the Singular Stratification
  14. Poles and Zeros of the Zeta Function $Z(\mathbb{P}_{\mathbf{w}}^{n}, t)$
  15. Riemann Hypothesis for $Z(\mathbb{P}_{\mathbf{w}}^{n}, t)$
  16. ...and 3 more sections

Key Result

Proposition 2.1

Under the standard interpretation of def:naive_strict (as def:standard_rational), the three notions of $\mathbb{F}_q$-rational points on $\mathbb{P}_\mathbf{w}^n$ are equivalent:

Theorems & Definitions (64)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Example 2.5
  • Lemma 2.3
  • ...and 54 more