Geometric zeta functions for higher rank p-adic groups

TL;DR

This paper proves the rationality of a multi-variable zeta function associated with p-adic groups acting on their Bruhat-Tits buildings, linking it to geometric zeta functions through specialization.

Contribution

It introduces a higher rank Lefschetz formula for p-adic groups and establishes the rationality of the associated multi-variable zeta functions.

Findings

Proves rationality of several-variable zeta functions

Relates multi-variable zeta functions to geometric zeta functions

Uses higher rank Lefschetz formula for 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