On the analogy between Arithmetic Geometry and foliated spaces

Christopher Deninger has developed an infinite dimensional cohomological formalism which allows to prove the expected properties of the arithmetical Zeta functions (including the Riemann Zeta function). These cohomologies are (in general) not yet constructed. Deninger has argued that these cohomologies might be constructed as leafwise cohomologies of suitable foliated spaces. We shall review some recent results which support this hope. 1 – Introduction Christopher Deninger’s approach to the study of arithmetic zeta functions proceeds in two steps. In the first step, he postulates the existence of infinite dimensional cohomology groups satisfying some “natural properties”. From these data, he has elaborated a formalism allowing him to prove the expected properties for the arithmetic zeta functions: functional equation, conjectures of Artin, Beilinson, Riemann . . . etc. There it is crucial to interpret the so called explicit formulae for the arithmetic zeta function as a Lefschetz trace formula. The second step consists in constructing these cohomologies. Deninger has given some hope that these cohomologies might be constructed as leafwise cohomologies of suitable foliated spaces. Very little is known in this direction

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