发表机构
Institute of Mathematics and Statistics, Rio de Janeiro State University; Department of Mathematics and Statistics, Oakland University（里约热内卢州立大学数学与统计研究所; 奥克兰大学数学与统计系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明加权超曲面的规范堆栈zeta函数具有有限扇区分解，给出特殊值的有理性与函数方程，并建立迷向熵的Lang–Weil渐近，将函数域高度等同于稳定堆栈高度。
AI 中文摘要
设 \\(X\subset \mathbb P^n_{\mathbf w}\\) 为 \\(\mathbb F_q\\) 上的加权超曲面，其关联堆栈为 \\(\mathfrak X\\)。对 \\(\mathbf x\in X(\mathbb F_{q^r})\\)，令 \\(g_{\mathbf x}(r)=\gcd(k_{S(\mathbf x)},q^r-1)\\)，其中 \\(k_S=\gcd(w_i:i\in S)\\)，并定义 \\(\Theta_r(s)=\sum_{\mathbf x}g_{\mathbf x}(r)^{1-s}\\) 与 \\(Z_H^{\mathrm{can}}(\mathfrak X,s;t)=\exp(\sum_{r\ge1}\Theta_r(s)t^r/r)\\)。我们证明了有限扇区分解 \\[ 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}, \\] 其中 \\(X^{(e)}\\) 为闭迷向扇区，\\(o_e=\operatorname{ord}_e(q)\\)（当 \\(e>1\\)）且 \\(o_1=1\\)，\\(J_{1-s}\\) 为Jordan totient函数。因此，特殊化 \\(s=1,0,-1,-2,\ldots\\) 在 \\(t\\) 中是有理的：\\(s=1\\) 给出粗空间的Hasse--Weil zeta函数，\\(s=0\\) 给出扭转zeta函数，\\(s=1-k\\) 给出 \\(k\\) 重惯性堆栈的质量。当非空扇区自对偶且等维时，该分解产生函数方程；若它们还纯且几何不可约，则对每个整数 \\(s\le0\\)，等维性是必要的。对于次数为 \\(D\\) 且 \\(p\nmid D\\) 的加权对角超曲面，有限Fermat覆盖给出纯性与自对偶性。我们还证明了迷向熵的Lang--Weil渐近，其周期首项由迷向周期和Frobenius轨道控制，并将函数域高度与Ellenberg--Satriano--Zureick-Brown的稳定堆栈高度等同，而基于次数的高度仍是坐标复杂度统计量。
英文摘要
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.
