Noncommutative geometry, dynamics, and ∞-adic Arakelov geometry

Abstract

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 consider derived (cohomological) spectral data $A, H^{\cdot}(X^*), \Phi$, where the algebra is obtained from the SL (2, ℝ) action on the cohomology of the cone, induced by the presence of a polarized Lefschetz module structure, and its restriction to the group ring of a Fuchsian Schottky group. In this setting we recover the alternating product of the archimedean factors from a zeta function of a spectral triple. Then, we introduce a different construction, which is related to Manin’s description of the dual graph of the fiber at infinity. We provide a geometric model for the dual graph as the mapping torus of a dynamical system T on a Cantor set. We consider a noncommutative space which describes the action of the Schottky group on its limit set and parameterizes the ‘‘components of the closed fiber at infinity’’. This can be identified with a Cuntz-Krieger algebra ${\mathcal O}_A$ associated to a subshift of finite type. We construct a spectral triple for this noncommutative space, via a representation on the cochains of a ‘‘dynamical cohomology’’, defined in terms of the tangle of bounded geodesics in the handlebody. In both constructions presented in the paper, the Dirac operator agrees with the grading operator Φ that represents the ‘‘logarithm of a Frobenius-type operator’’ on the archimedean cohomology. In fact, the archimedean cohomology embeds in the dynamical cohomology, compatibly with the action of a real Frobenius $\bar F_\infty,$ so that the local factor can again be recovered from these data. 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. Moreover, the ‘‘reduction mod infinity’’ is described in terms of the homotopy quotient associated to the noncommutative space ${\mathcal O}_A$ and the μ-map of Baum-Connes. The geometric model of the dual graph can also be described as a homotopy quotient.