This paper explores analogies between the Weil proof of the Riemann hypothesis for function fields and the geometry of the adeles class space, which is the noncommutative space underlying Connes spectral realization of the zeros of the Riemann zeta f
unction. We consider the cyclic homology of the cokernel (in the abelian category of cyclic modules) of the ``restriction map defined by the inclusion of the ideles class group of a global field in the noncommutative adeles class space. Weils explicit formula can then be formulated as a Lefschetz trace formula for the induced action of the ideles class group on this cohomology. In this formulation the Riemann hypothesis becomes equivalent to the positivity of the relevant trace pairing. This result suggests a possible dictionary between the steps in the Weil proof and corresponding notions involving the noncommutative geometry of the adeles class space, with good working notions of correspondences, degree and codegree etc. In particular, we construct an analog for number fields of the algebraic points of the curve for function fields, realized here as classical points (low temperature KMS states) of quantum statistical mechanical systems naturally associated to the periodic orbits of the action of the ideles class group, that is, to the noncommutative spaces on which the geometric side of the trace formula is supported.
In this paper we construct a noncommutative space of ``pointed Drinfeld modules that generalizes to the case of function fields the noncommutative spaces of commensurability classes of Q-lattices. It extends the usual moduli spaces of Drinfeld module
s to possibly degenerate level structures. In the second part of the paper we develop some notions of quantum statistical mechanics in positive characteristic and we show that, in the case of Drinfeld modules of rank one, there is a natural time evolution on the associated noncommutative space, which is closely related to the positive characteristic L-functions introduced by Goss. The points of the usual moduli space of Drinfeld modules define KMS functionals for this time evolution. We also show that the scaling action on the dual system is induced by a Frobenius action, up to a Wick rotation to imaginary time.
This is the text of a series of five lectures given by the author at the Second Annual Spring Institute on Noncommutative Geometry and Operator Algebras held at Vanderbilt University in May 2004. It is meant as an overview of recent results illustrat
ing the interplay between noncommutative geometry and arithmetic geometry/number theory.
We construct spectral triples associated to Schottky--Mumford curves, in such a way that the local Euler factor can be recovered from the zeta functions of such spectral triples. We propose a way of extending this construction to the case where the curve is not k-split degenerate.
This paper consists of variations upon the theme of limiting modular symbols. Topics covered are: an expression of limiting modular symbols as Birkhoff averages on level sets of the Lyapunov exponent of the shift of the continued fraction, a vanishin
g theorem depending on the spectral properties of a generalized Gauss-Kuzmin operator, the construction of certain non-trivial homology classes associated to non-closed geodesics on modular curves, certain Selberg zeta functions and C^* algebras related to shift invariant sets.
