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 $Γ\backslash\mathscr{B}$, 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 $Q=|\mathcal{O}_D/\mathfrak{p}_D|$.

Disclosure

“is formal and depends only on the finite cochain complex of pointed facets. Applying it to BD,Γ proves the displayed identity. □ Acknowledgments The authors acknowledge the use of ChatGPT for editorial assistance in improving the exposition and organization of this manuscript. References [AB08] Peter Abramenko and Kenneth S. Brown, Buildings, Graduate Texts in Mathema”

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Brainstorming or outlining

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2

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2026-08-23T19:03:26.146951Z

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