Geometric zeta functions for higher rank $p$-adic groups

Abstract

The higher rank Lefschetz formula for $p$-adic groups is used to prove rationality of a several-variable zeta function attached to the action of a $p$-adic group on its Bruhat–Tits building. By specializing to certain lines one gets one-variable zeta functions, which then can be related to geometrically defined zeta functions.