Quantum statistical mechanics over function fields

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 modules 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.Comment: 28 pages, LaTeX; v2: last section expande

Spectral triples from Mumford curves

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.Comment: 25 pages, LaTeX, 4 eps figures (v4: to appear in IMRN

Archimedean cohomology revisited

Archimedean cohomology provides a cohomological interpretation for the calculation of the local L-factors at archimedean places as zeta regularized determinant of a log of Frobenius. In this paper we investigate further the properties of the Lefschetz and log of monodromy operators on this cohomology. We use the Connes-Kreimer formalism of renormalization to obtain a fuchsian connection whose residue is the log of the monodromy. We also present a dictionary of analogies between the geometry of a tubular neighborhood of the ``fiber at arithmetic infinity'' of an arithmetic variety and the complex of nearby cycles in the geometry of a degeneration over a disk, and we recall Deninger's approach to the archimedean cohomology through an interpretation as global sections of a analytic Rees sheaf. We show that action of the Lefschetz, the log of monodromy and the log of Frobenius on the archimedean cohomology combine to determine a spectral triple in the sense of Connes. The archimedean part of the Hasse-Weil L-function appears as a zeta function of this spectral triple. We also outline some formal analogies between this cohomological theory at arithmetic infinity and Givental's homological geometry on loop spaces.Comment: 28 pages LaTeX 3 eps figure

The Cyclic and Epicyclic Sites

We determine the points of the epicyclic topos which plays a key role in the geometric encoding of cyclic homology and the lambda operations. We show that the category of points of the epicyclic topos is equivalent to projective geometry in characteristic one over algebraic extensions of the infinite semifield of max-plus integers. An object of this category is a pair of an algebraic extension of the semifield and an archimedean semimodule over this extension. The morphisms are projective classes of semilinear maps between semimodules. The epicyclic topos sits over the arithmetic topos which we recently introduced and the fibers of the associated geometric morphism correspond to the cyclic site. In two appendices we review the role of the cyclic and epicyclic toposes as the geometric structures supporting cyclic homology and the lambda operations.Comment: 35 pages, 5 figure

From monoids to hyperstructures: in search of an absolute arithmetic

We show that the trace formula interpretation of the explicit formulas expresses the counting function N(q) of the hypothetical curve C associated to the Riemann zeta function, as an intersection number involving the scaling action on the adele class space. Then, we discuss the algebraic structure of the adele class space both as a monoid and as a hyperring. We construct an extension R^{convex} of the hyperfield S of signs, which is the hyperfield analogue of the semifield R_+^{max} of tropical geometry, admitting a one parameter group of automorphisms fixing S. Finally, we develop function theory over Spec(S) and we show how to recover the field of real numbers from a purely algebraic construction, as the function theory over Spec(S).Comment: 43 pages, 1 figur