Introduction

For the arithmetic study of varieties over finite fields powerful cohomological methods are available which in particular shed much light on the nature of the corresponding zeta functions. For algebraic schemes over spec ℤ and in particular for the Riemann zeta function no cohomology theory has yet been developed that could serve similar purposes. For a long time it had even been a mystery how the formalism of such a theory could look like. This was clarified in [D1]. However until now the conjectured cohomology has not been constructed.

There is a simple class of dynamical systems on foliated manifolds whose reduced leafwise cohomology has several of the expected structural properties of the desired cohomology for algebraic schemes. In this analogy, the case where the foliation has a dense leaf corresponds to the case where the algebraic scheme is flat over spec ℤ e.g. to spec ℤ itself. In this situation the foliation cohomology which in general is infinite dimensional is not of a topological but instead of a very analytic nature. This can also be seen from its description in terms of global differential forms which are harmonic along the leaves. An optimistic guess would be that for arithmetic schemes χ there exist foliated dynamical systems X whose reduced leafwise cohomology gives the desired cohomology of χ. If χ is an elliptic curve over a finite field this is indeed the case with X a generalized solenoid, not a manifold, [D3].

We illustrate this philosophy by comparing the “explicit formulas” in analytic number theory to a transversal index theorem.