ar X iv : d g - ga / 9 70 10 08 v 3 2 1 A pr 1 99 7 Geometric zeta - functions on p - adic groups ∗

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 PSL2. 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 PSL2 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].