Zeta functions of PGL_n over non-Archimedean local fields

2026/07/23 by Ming-Hsuan Kang, Jiu-Kang Yu
#math.NT

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Abstract

Let \mathscrB be the Bruhat--Tits building of PGL_n(F), where F is a non-Archimedean local field. We introduce geometric k-geodesics in \mathscrB 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\mathscrB, 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 L²(Γ\backslash PGL_n(F)): (1-uⁿ)^χ(Γ\backslash\mathscrB)L(Γ,q^(n-1)/2u)=∏_k=1^n-1 Z_k^ε(Γ\backslash\mathscrB,u)^(-1)^k+1. This gives a uniform Ihara-type identity for all PGL_n. We also extend the construction and the identity to PGL_n(D), where D is a central division algebra over F; in that setting the residue parameter is Q=|O_D/\mathfrakp_D|.

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