Orbit counts and twist zeta functions of weighted projective stacks over finite fields

Abstract

Let ${\mathbb F}_q$ be a finite field and $\mathbf{w}=(w_0,\dots,w_n)$ a vector of positive integer weights. Several finite-field counts attached to the weighted projective space $\mathbb{P}^n_\mathbf{w}$ are easily conflated: the coarse rational-point count and the stacky mass are both weight-independent, whereas the number $A_\mathbf{w}(q)$ of ${\mathbb F}_q^\times$-orbits on nonzero ${\mathbb F}_q$-representatives for the weighted action depends on the weights. We prove the closed formula \[ A_\mathbf{w}(q)=\sum_{\emptyset\ne S\subseteq\{0,\dots,n\}}(q-1)^{|S|-1}\gcd(k_S,q-1), \qquad k_S=\gcd\{w_i:i\in S\}, \] and identify $A_\mathbf{w}(q)$ intrinsically as the number of ${\mathbb F}_q$-isomorphism classes of the weighted projective stack $\mathcal{P}_\mathbf{w}=[({\mathbb A}^{n+1}\setminus\{0\})/\mathbb{G}_m]$ -- equivalently, the number of ${\mathbb F}_q$-twists lying over the coarse points -- the discrepancy from the coarse count being governed by the Kummer groups ${\mathbb F}_q^\times/({\mathbb F}_q^\times)^{k_S}$. We read the behaviour of $A_\mathbf{w}$ under reduction of the weight vector through the ${\mathbb F}_q$-cohomology of the associated $\boldsymbolμ_d$-gerbe, and prove that the twist zeta function $Z_{\mathrm{tw}}(\mathcal{P}_\mathbf{w},t)=\exp\bigl(\sum_{r\ge1}A_\mathbf{w}(q^r)t^r/r\bigr)$ is rational with multiplicity spectrum independent of $q$, admitting a single global functional equation precisely when the weights share a common prime-to-$p$ part -- for reduced $\mathbf{w}$, precisely when the twist theory is trivial. For weighted diagonal hypersurfaces and same-degree pairs in the split regime, we compute the twist zeta function of the substack explicitly, with reciprocal roots given by Gauss sums and, for intersections, by Frobenius eigenvalues of superelliptic curves.