AI 中文总结
该研究针对 $\text{PGL}_d$ 的标准算术商，基于建筑理论计算顶点体积，引入同伦不变归一性格极小值高度，确定其可积阈值与尖点尾估计，推导高度泽塔函数的收敛域与解析延拓性质，明确计算了 $d=3,4,5$ 时的有理函数。
AI 中文摘要
我们研究与 $\boldsymbol{\text{PGL}}_d(\boldsymbol{\text{F}}_q(\boldsymbol{(t^{-1})}))$ 关联的仿射 Bruhat-Tits 建筑的标准非均匀算术商，其中 Haar 测度的归一化使得极大紧子群的体积为 1。我们首先以闭合乘积形式计算其顶点体积，证明完全基于建筑理论：顶点由优势扇区参数化，其稳定子群被精确计数，对块组合的求和通过割集递推进行计算。在同一商空间上，我们引入一个同伦不变的归一性格极小值高度 $\boldsymbol{\text{\textit{α}}}$，确定其精确可积阈值，证明当且仅当 $0<r<d$ 时 $\boldsymbol{\text{\textit{α}}}$ 属于 $L^r$，并建立阶为 $\boldsymbol{T^{-d}}$ 的精确尖点尾估计。相关的正矩高度泽塔函数（等价于尖点高度分布的 Mellin 变换）恰好收敛于半平面 $\boldsymbol{\text{Re}(s)<d}$，它作为 $\boldsymbol{q^{s/d}}$ 的有理函数允许亚纯延拓，且在 $\boldsymbol{s=d}$ 处有一个简单极点，带有显式临界系数。我们还对 $\boldsymbol{d=3,4,5}$ 明确计算了所得的有理函数，因此同一优势扇区坐标同时控制体积、尖点衰减和高度泽塔函数的解析结构。
英文摘要
We study the standard nonuniform arithmetic quotient of the affine Bruhat--Tits building attached to $\operatorname{PGL}_d(\mathbb F_q(\!(t^{-1})\!))$, with Haar measure normalized so that a maximal compact subgroup has volume one. We first compute its vertex volume in closed product form. The proof is entirely building-theoretic: vertices are parametrized by a dominant sector, their stabilizers are counted exactly, and the resulting sum over block compositions is evaluated by a cut-set recursion. On the same quotient, we introduce a homothety-invariant normalized lattice-minima height $α$. We determine its exact integrability threshold, proving that $α$ belongs to $L^r$ precisely for $0<r<d$, and establish a sharp cusp-tail estimate of order $T^{-d}$. The associated positive-moment height zeta function, equivalently the Mellin transform of the cusp-height distribution, converges exactly in the half-plane $\operatorname{Re}(s)<d$. It admits a meromorphic continuation as a rational function of $q^{s/d}$ and has a simple pole at $s=d$, with an explicit critical coefficient. We also compute the resulting rational functions explicitly for $d=3,4,5$. Thus the same dominant-sector coordinates simultaneously control volume, cusp decay, and the analytic structure of the height zeta function.
Comments22 pages, comments welcome!
