Geometric Zeta Functions, L-Theory, and Compact Shimura Manifolds

INTRODUCTION 4 Introduction Zeta functions encoding geometric information such as zeta functions of algebraic varieties over finite fields or zeta functions of finite graphs will loosely be called geometric zeta functions in the sequel. Sometimes the geometric situation gives one tools at hand to prove analytical continuation, functional equation and an adapted form of the Riemann hypothesis. This works as follows: The geometrical data is related to the fixed point set of an operator, this fixed point set is considered as the local data. There is a Lefschetz formula relating these to the action of this operator on a global object such as a suitable cohomology theory, thus expressing the zeta function by determinants of the operator. This is what we will call a determinant formula. This could have been the guiding idea when Selberg in 1956 [Sel] first defined a geometric zeta function for compact Riemannian surfaces by now known as the Selberg zeta function. To play the rôle of a Lefschetz formula Selberg designed his by now famous trace formula which soon was generalized to higher dimensional Shimura manifolds, i.e. quotients of symmetric spaces. The generalization of the zeta function took somewhat more time. In 1977 R. Gangolli [Gang] defined a zeta functions for all compact rank one Shimura man-ifolds. In 1985 M. Wakayama defined twisted versions of these [Wak]. Up to this point all the work was formulated in terms of the harmonic analysis not obeying the above philosophy and there was no such a thing as a determinant formula which would have given a more comprehensive way to formulate results and deeper insights into the connection between zeta functions and the topology of Shimura manifolds. Since the generalized Selberg zeta functions have infinitely many zeroes it was clear that a determinant formula would require a notion of a determinant of infinite rank operators. The latter was provided by the Ray-Singer regularized determinant [RS-AT]. In 1986 D. Fried [Fr] proved a determinant formula for the real hyperbolic spaces which are special symmetric spaces of rank one. Fried further showed that certain products of the generalized Selberg functions in these cases give the Ruelle zeta function of the geodesic flow. D. Ruelle had defined the zeta function carrying his name ten years earlier in a more general context [Rue] where he was able to show the existence of a meromorphic continuation but not very much more than that. …