I‑AIMS Institute for Artificial Intelligence
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Stacky Heights and Zeta Functions of Weighted Hypersurfaces

Sajad Salami, Tanush Shaska

I-AIMS preprint sh-112 · 2026

Let X ⊂ ℙⁿ_w be a weighted hypersurface over 𝔽_q, and let 𝒳 be the associated weighted projective stack. If x ∈ 𝒳(𝔽_(qʳ)) has support S(x), put gₓ(r) = gcd(k_(S(x)), qʳ − 1), where k_S = gcd(wᵢ: i ∈ S). We study the coefficient functions Θᵣ(s) = ∑_(x ∈ 𝒳(𝔽_(qʳ))) gₓ(r)¹⁻ˢ and the two-variable series Zᶜᵃⁿ_H(𝒳, s; t) = exp(∑_(r ≥ 1) Θᵣ(s) tʳ / r). Our main result is a finite sector factorization Zᶜᵃⁿ_H(𝒳, s; t) = ∏_(e ∈ E_w) Z(X(e) ⊗_(𝔽_q) 𝔽_(q^(oₑ)), t^(oₑ))^(J₁₋ₛ(e)/oₑ), where X(e) is the closed isotropy sector cut out by the coordinates whose weights are not divisible by e, oₑ = ordₑ(q) for e > 1 with o₁:= 1, and J₁₋ₛ is the Jordan totient function. Hence the specializations s = 1, 0, −1, −2, … are rational in t; at s = 1 one recovers the Hasse–Weil zeta function of the coarse space, at s = 0 the twist zeta function, and at s = 1 − k the coefficients are the masses of the k-fold inertia stack. The same factorization yields a functional equation whenever the nonempty sectors are self-dual and equidimensional. Under the additional hypotheses that the sectors are pure and geometrically irreducible, equidimensionality is also necessary for every integral specialization s ≤ 0, giving an exact criterion for the twist zeta function in this class. For weighted diagonal hypersurfaces of degree D with p ∤ D, the required purity and self-duality follow from a finite Fermat cover. We also prove a Lang–Weil asymptotic for the entropy of the isotropy stratification, with the periodic leading coefficient determined jointly by isotropy periods and Frobenius orbits of geometric components. Finally, the function-field height is identified with the stable stack height of Ellenberg–Satriano–Zureick-Brown, while the degree-based height is kept as a separate coordinate-complexity statistic.

MSC classes: Primary 11G25, 14G15; Secondary 14A20, 11G50, 14G10, 94A17

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