Triplets Spectraux En Géométrie D'arakelov Spectral Triples in Arakelov Geometry Abridged English Version

In this note, we use Connes’ theory of spectral triples to provide a connection between Manin’s model of the dual graph of the fiber at infinity of an Arakelov surface and the cohomology of the mapping cone of the local monodromy. Abridged English version In Arakelov theory a completion of an arithmetic surface is achieved by enlarging the group of divisors by formal linear combinations of the “closed fibers at infinity”. Manin in [8] described the dual graph of any such closed fiber in terms of an infinite tangle of bounded geodesics in a hyperbolic handlebody XΓ = Γ\H3, uniformized by a Schottky group Γ ⊂ PSL(2,C). In this note we consider arithmetic surfaces over the ring of integers in a number field, with fibers of genus g ≥ 2. We use Connes’ theory of spectral triples to relate the hyperbolic geometry of the handlebody to the cohomology of the cone of the local monodromyN at arithmetic infinity as introduced in [4]. First, we construct a spectral triple, where the non–commutative space is given by the reduced C–algebra of the Schottky group acting on the cohomology of the cone via a representation induced by the presence of a polarized Lefschetz module structure. In this setting we recover the alternating product of the archimedean factors from a zeta function of the spectral triple. Then, we introduce a second spectral triple, which is related to Manin’s description of the dual graph of the fiber at infinity. Here the non–commutative space is a C–algebra representing the “reduction mod infinity” and acting on a “dynamical homology and cohomology” pair, defined in terms of the bounded geodesics in the handlebody and of a dynamical system T . The operator Φ, that represents the “logarithm of a Frobenius–type operator” on the archimedean cohomology of [7], gives the Dirac operator on these spectral triples. We show that the archimedean cohomology embeds in the dynamical cohomology, compatibly with the action of a real Frobenius F̄∞, so that the duality isomorphism on the cohomology of the cone of N corresponds to the pairing of dynamical homology and cohomology. A detailed version of the results presented in this note is contained in [5]. Partiellement supportée par la bourse du NSERC numéro 72016789 Partiellement supportée par la bourse Sofja Kovalevskaja de l’Humboldt Foundation