AI 中文总结
研究非阿基米德局部域上\(\mathrm{PGL}_n\)的泽塔函数，通过引入几何与组合\(k -\)测地线并证明其重合，在商空间上定义相关函数，得到泽塔函数交错积与非分歧\(L -\)函数的等同关系，并扩展到\(\mathrm{PGL}_n(D)\)。
AI 中文摘要
设\(\mathscr{B}\)是\(\mathrm{PGL}_n(F)\)的布鲁哈特 - 蒂茨建筑，其中\(F\)是非阿基米德局部域。通过CAT(0)凸性引入\(\mathscr{B}\)中的几何\(k -\)测地线，通过在有向\(k -\)面的局部后继关系引入组合\(k -\)测地线，并证明二者重合。这使得能在商空间\(\Gamma\backslash\mathscr{B}\)上使用局部组合定义。当\(\Gamma\)满足一定条件时，原始闭\(k -\)测地线定义泽塔函数\(Z_k\)及其\(\epsilon -\)扭曲变体\(Z_k^\epsilon\)。主要结果是将这些泽塔函数的交错积与\(L^2(\Gamma\backslash \mathrm{PGL}_n(F))\)的非分歧\(L -\)函数等同起来，还将构造和恒等式扩展到\(\mathrm{PGL}_n(D)\)。
英文摘要
Let $\mathscr{B}$ be the Bruhat--Tits building of $\mathrm{PGL}_n(F)$, where $F$ is a non-Archimedean local field. We introduce geometric $k$-geodesics in $\mathscr{B}$ by means of CAT(0) convexity and combinatorial $k$-geodesics by a local successor relation on pointed $k$-facets. We prove that the two notions coincide. This allows us to use the local combinatorial definition on quotients $Γ\backslash\mathscr{B}$, without referring to the universal covering. When $Γ$ is discrete, torsion-free, cocompact, and type-preserving, the primitive closed $k$-geodesics define zeta functions $Z_k$ and their $ε$-twisted variants $Z_k^ε$. Our main result identifies an alternating product of these zeta functions with the unramified $L$-function of $L^2(Γ\backslash \mathrm{PGL}_n(F))$: $(1-u^n)^{χ(Γ\backslash\mathscr{B})}L(Γ,q^{(n-1)/2}u)=\prod_{k=1}^{n-1} Z_k^ε(Γ\backslash\mathscr{B},u)^{(-1)^{k+1}}$. This gives a uniform Ihara-type identity for all $\mathrm{PGL}_n$. We also extend the construction and the identity to $\mathrm{PGL}_n(D)$, where $D$ is a central division algebra over $F$; in that setting the residue parameter is $Q=|\mathcal{O}_D/\mathfrak{p}_D|$.
Comments47 pages
