Zeta functions of ${\mathbb F}_1$-buildings

Abstract

The analogue of the Bruhat–Tits building of a $p$-adic group in F1-geometry is a single apartment. In this setting, the trace formula gives rise to a several variable zeta function analogously to the $p$-adic case. The analogy carries on to the fact that the restriction to certain lines yield zeta functions which are defined in geometrical terms. Also, the classical formula of Ihara has an analogue in this setting.