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

Source abstract

Let $X\subset \mathbb P^n_{\mathbf w}$ be a weighted hypersurface over $\mathbb F_q$, with associated stack $\mathfrak X$. For $\mathbf x\in X(\mathbb F_{q^r})$, let $g_{\mathbf x}(r)=\gcd(k_{S(\mathbf x)},q^r-1)$, where $k_S=\gcd(w_i:i\in S)$, and define $Θ_r(s)=\sum_{\mathbf x}g_{\mathbf x}(r)^{1-s}$ and $Z_H^{\mathrm{can}}(\mathfrak X,s;t)=\exp(\sum_{r\ge1}Θ_r(s)t^r/r)$. We prove the finite sector factorization $$Z_H^{\mathrm{can}}(\mathfrak X,s;t)=\prod_{e\in E_{\mathbf w}} Z(X^{(e)}_{\mathbb F_{q^{o_e}}},t^{o_e})^{J_{1-s}(e)/o_e},$$ where $X^{(e)}$ is the closed isotropy sector, $o_e=\operatorname{ord}_e(q)$ for $e>1$ and $o_1=1$, and $J_{1-s}$ is the Jordan totient function. Hence the specializations $s=1,0,-1,-2,\ldots$ 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 $s\le0$. For weighted diagonal hypersurfaces of degree $D$ with $p\nmid D$, 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.