Stacky Heights, Entropy, and Zeta Functions of Weighted Hypersurfaces over Finite Fields

Abstract

Let X⊂ ℙⁿ_w be a weighted hypersurface over 𝔽_q, with associated stack X. For x∈ X(𝔽_q^r), let g_x(r)=gcd(k_S(x),q^r−1), where k_S=gcd(w_i:i∈ S), and define Θ_r(s)=∑_xg_x(r)^1−s and Z_H^can(X,s;t)=exp(∑_r≥1Θ_r(s)t^r/r). We prove the finite sector factorization Z_H^can(X,s;t)=∏_{e∈ E_w} Z(X^(e)_{𝔽_{q^o_e}},t^o_e)^{J_1−s(e)/o_e}, where X^(e) is the closed isotropy sector, o_e=ord_e(q) for e>1 and o₁=1, and J_1−s is the Jordan totient function. Hence the specializations s=1,0,−1,−2,… are rational in t: s=1 gives the Hasse--Weil zeta function of the coarse space, s=0 the twist zeta function, and s=1−k the masses of the k-fold inertia stack. The factorization yields a functional equation when the nonempty sectors are self-dual and equidimensional; if they are also pure and geometrically irreducible, equidimensionality is necessary for every integral s≤0. For weighted diagonal hypersurfaces of degree D with pnmid D, a finite Fermat cover gives purity and self-duality. We also prove a Lang--Weil asymptotic for isotropy entropy, with periodic leading term governed by isotropy periods and Frobenius orbits, and identify the function-field height with the stable stack height of Ellenberg--Satriano--Zureick-Brown, while the degree-based height remains a coordinate-complexity statistic.

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BibTeX RIS

BibTeX

@misc{salami2026stacky,
  title = {{Stacky Heights, Entropy, and Zeta Functions of Weighted Hypersurfaces over Finite Fields}},
  author = {S. Salami and T. Shaska},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33162},
  eprint = {2609.33162},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33162},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Stacky Heights, Entropy, and Zeta Functions of Weighted Hypersurfaces over Finite Fields
AU  - S. Salami
AU  - T. Shaska
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.33162
UR  - https://arxiv.org/abs/2609.33162
AB  - Let X⊂ ℙ^n_w be a weighted hypersurface over 𝔽_q, with associated stack X. For x∈ X(𝔽_q^r), let g_x(r)=gcd(k_S(x),q^r−1), where k_S=gcd(w_i:i∈ S), and define Θ_r(s)=∑_xg_x(r)^1−s and Z_H^can(X,s;t)=exp(∑_r≥1Θ_r(s)t^r/r). We prove the finite sector factorization Z_H^can(X,s;t)=∏_e∈ E_w Z(X^(e)_𝔽_q^o_e,t^o_e)^J_1−s(e)/o_e, where X^(e) is the closed isotropy sector, o_e=ord_e(q) for e>1 and o_1=1, and J_1−s is the Jordan totient function. Hence the specializations s=1,0,−1,−2,… are rational in t: s=1 gives the Hasse--Weil zeta function of the coarse space, s=0 the twist zeta function, and s=1−k the masses of the k-fold inertia stack. The factorization yields a functional equation when the nonempty sectors are self-dual and equidimensional; if they are also pure and geometrically irreducible, equidimensionality is necessary for every integral s≤0. For weighted diagonal hypersurfaces of degree D with pnmid D, a finite Fermat cover gives purity and self-duality. We also prove a Lang--Weil asymptotic for isotropy entropy, with periodic leading term governed by isotropy periods and Frobenius orbits, and identify the function-field height with the stable stack height of Ellenberg--Satriano--Zureick-Brown, while the degree-based height remains a coordinate-complexity statistic.
ER  -

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