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

Sajad Salami, Tanush Shaska

I-AIMS preprint sh-87 · 2025

Let 𝔽_q be a finite field and w = (w₀, …, wₙ) a vector of positive integer weights. Several finite-field counts attached to the weighted projective space ℙⁿ_w are easily conflated: the coarse rational-point count and the stacky mass are both weight-independent, whereas the number A_w(q) of 𝔽_q^×-orbits on nonzero 𝔽_q-representatives for the weighted action depends on the weights. We prove the closed formula A_w(q) = ∑_(∅ ≠ S ⊆ {0, …, n}) (q−1)^(|S|−1) gcd(k_S, q−1), k_S = gcd{wᵢ: i ∈ S}, and identify A_w(q) intrinsically as the number of 𝔽_q-isomorphism classes of the weighted projective stack 𝒫_w = [(𝔸ⁿ⁺¹ ∖ {0})/𝔾ₘ]—equivalently, the number of 𝔽_q-twists lying over the coarse points—the discrepancy from the coarse count being governed by the Kummer groups 𝔽_q^× / (𝔽_q^×)^(k_S). We read the behaviour of A_w under reduction of the weight vector through the 𝔽_q-cohomology of the associated μ_d-gerbe, and prove that the twist zeta function Z_(tw)(𝒫_w, t) = exp(∑_(r ≥ 1) A_w(qʳ) tʳ / r) 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 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.

MSC classes: Primary 14G15, 14A20; Secondary 14M25, 14G10, 11G25

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