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

Abstract

We generalize the theory of p-adic geometric zeta functions of Y. Ihara and K. Hashimoto to the higher rank case. We give the proof of rationality of the zeta function and the connection of the divisor to group cohomology, i.e. the p-adic analogue of the Patterson conjecture.Introduction. In [13] and [14] Y. Ihara defined geometric zeta functions for the group P SL 2 . This was the p-adic counterpart of the famous zeta functions of Selberg for Riemannian surfaces. Using his results on the structure of discrete subgroups of P SL 2 of a p-adic field he proved, among other things, the rationality of these zeta functions. By means of the geometric interpretation on the building, K. Hashimoto [10], [11], [12] extended this to arbitrary rank one groups. In the present paper we generalize Ihara's theory to the higher rank case. Our method uses harmonic analysis and is insofar not geometrical. We will provide geometric interpretations in subsequent work. We prove rationality of the zeta functions and show that the poles and zeroes are related to the cohomology of the discrete group, i.e. the p-adic counterpart of the Patterson-conjecture [6].

Building lattices and zeta functions

“…In the latter case a Lefschetz formula has been developed [1,9,11,13,20,21], which expresses geometrical data of the geodesic flow and its monodromy in terms of Lie algebra cohomology, or more general, foliation cohomology. This has been transferred to the case of p-adic groups in [10] and applied in [15]. The presentation in both papers is focused on the cohomological approach using the theory of reductive linear algebraic groups.…”

Benjamini–Schramm convergence and zeta functions

“…1, extending a previous notion by the author [9]. Suitable zeta functions are present in the literature, at least for Lie groups [4,11,22] or p-adic groups [7,14]. A general definition for arbitrary locally compact groups is still lacking.…”

High Dimensional Expanders

“…An interesting direction of research is to try to associate to high dimensional complexes suitable "zeta functions" with the hope that also in this context the Ramanujaness of the complex can be expressed via the RH. For this direction or research -see [Sto06], [KaLi14], [DK14], [KLW10], [Kan16], [Kam17b] and [LLP17].…”