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Noncommutative geometry, dynamics, and \(\infty\)-adic Arakelov geometry. (English) Zbl 1112.14023

Let \(K\) be a number field and \(\mathcal{O}_K\) be the ring of integers. The authors of the article under review consider an arithmetic surface \(X_{\mathcal{O}_K}\) over \(\text{Spec}\,(\mathcal{O}_K)\) (the model of a smooth, algebraic curve \(X\) over \(K\)), with fibers of genus \(g\geq 2\). The completion of this surface is achieved in Arakelov theory by adding to \(\text{Spec}\,(\mathcal{O}_K)\) the archimedean places corresponding to the embeddings \(K\hookrightarrow\mathbb{C}\). The formal real combinations of the closed vertical fibers at the infinity are taken into consideration when defining the Arakelov divisors on the completion of \(X_{\mathcal{O}_K}\).
In the first part of the paper the authors consider the formal construction of a cohomological theory developed by the first author [Compos. Math. 111, 323–358 (1998; Zbl 0932.14011)]. The Riemann surface \(X_{\mathbb{C}}\) supports a double complex \((K^{\cdot,\cdot},d',d'')\) endowed with a homomorphism \(N\). In this part of the paper the authors concentrate on the cohomology \(H^{\cdot}(X^*)\) of \(\text{Cone}\,(N)\). They consider the spectral triple \((A,H^{\cdot}(X^*),\Phi)\) defined by the bigraded Lefschetz module structure on the complex \((K^{\cdot},d=d'+d'')\). The algebra \(A\) is obtained from the action of SL\((2,\mathbb{R})\) on the cohomology of the cone, induced by the Lefschetz module structure, and its restriction to the group ring of a Fuchsian Schottky group. The operator \(\Phi\) determines the archimedian factors of C. Deninger [Invent. Math. 104, 245–261 (1991; Zbl 0739.14010)] and behaves like a Dirac operator. Deninger’s regularized determinants are interpreted in terms of an integration theory on the “noncommutative manifold” \((A,H^{\cdot}(X^*),\Phi)\).
The second construction presented in the paper is related to Manin’s description of the dual graph of the fiber at infinity of the arithmetic surface \(X_{\mathcal{O}_K}\), in terms of the infinite tangle of bounded geodesics in a hyperbolic handlebody endowed with a Schottky uniformization [Yu. I. Manin, Invent. Math. 104, 223–244 (1991; Zbl 0754.14014)]. The authors describe a model of the dual graph of the fiber at infinity, as the suspension flow of a dynamical system \(T\). They also define a dynamic cohomology (resp. homology) \(H^1_{\text{dyn}}\) (\(H_1^{\text{dyn}}\)) of the fiber at infinity, and these graded spaces have an involution that plays a role similar to the real Frobenius on the cohomological theories in the first part of the article. Theorem 5.7 relates the dynamical cohomology to the the archimedean cohomology. More precisely, the archimedean cohomology sits as a particular subspace of the dynamical one in a way that is compatible with the grading and the action of the real Frobenius. In Theorem 5.12 the authors reinterpret the duality isomorphism acting on \(\mathbb{H}^{\cdot}(\text{Cone}^{\cdot}(N))\) as induced by the pairing of the dynamical homology and cohomology.
The action of the Schottky group on its limit set is described using a Cuntz-Krieger algebra \(\mathcal{O}_A\) associated to a shift of finite type. A spectral triple for this noncommutative space is constructed (Theorem 6.6), via a representation on the cochains of the dynamical cohomology, and the local Euler factor at arithmetic infinity is recovered from this data (Theorem 6.8).
In section 7 the authors describe the analog at arithmetic infinity of the \(p\)-adic reduction map. This is realized in terms of a homotopy quotient of the space \(\mathcal{O}_A\) and the \(\mu\)-map of Baum-Connes. The geometric model of the dual graph ia also described as a homotopy quotient.
The last section of the paper contains some questions and directions for further investigation.

MSC:

14G40 Arithmetic varieties and schemes; Arakelov theory; heights
58B34 Noncommutative geometry (à la Connes)
46L55 Noncommutative dynamical systems
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