Non–commutative geometry, dynamics, and ∞–adic Arakelov geometry

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 described the dual graph of any such closed fiber in terms of an infinite tangle of bounded geodesics in a hyperbolic handlebody endowed with a Schottky uniformization. In this paper 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 Deninger’s Archimedean cohomology and the cohomology of the cone of the local monodromy N at arithmetic infinity as introduced by the first author of this paper. 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 . Morita equivalence plays an important role in the definition of the action. The operator Φ, that represents the “logarithm of a Frobenius–type operator” on the Archimedean cohomology, gives the Dirac operator on these spectral triples. 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. This suggests the existence of a duality between the monodromy N and the dynamical map 1− T . Partially supported by NSERC grant 72016789 Partially supported by Humboldt Foundation Sofja Kovalevskaja Award