Zeta functions of $\mathrm{PGL}_n$ over non-Archimedean local fields

Abstract

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 $\Gamma\backslash\mathscr{B}$, without referring to the universal covering. When $\Gamma$ is discrete, torsion-free, cocompact, and type-preserving, the primitive closed $k$-geodesics define zeta functions $Z_k$ and their $\epsilon$-twisted variants $Z_k^\epsilon$. Our main result identifies an alternating product of these zeta functions with the unramified $L$-function of $L^2(\Gamma\backslash \mathrm{PGL}_n(F))$: $(1-u^n)^{\chi(\Gamma\backslash\mathscr{B})}L(\Gamma,q^{(n-1)/2}u)=\prod_{k=1}^{n-1} Z_k^\epsilon(\Gamma\backslash\mathscr{B},u)^{(-1)^{k+1}}$. 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 $Q=|\mathcal{O}_D/\mathfrak{p}_D|$.