Spectral triples in Arakelov geometry
From the 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''. \textit{Yu. I. Manin} [Invent. Math. 104, 223-243 (1991; Zbl 0754.14014)]\ described the dual graph of any such closed fiber in terms of an infinite tangle of bounded geodesics in a hyperbolic handlebody \({\mathfrak X}_\Gamma= \Gamma \backslash \mathbb{H}^3\), uniformized by a Schottky group \(\Gamma\subset \text{PSL} (2,\mathbb{C})\). In this note we consider arithmetic surfaces over the ring of integers in a number field, with fibers of genus \(g\geq 2\). We use \textit{A. Connes}' theory [Lett. Math. Phys. 34, 203-238 (1995; Zbl 1042.46515)] of spectral triples to relate the hyperbolic geometry of the handlebody to the cohomology of the cone of the local monodromy \(N\) at arithmetic infinity as introduced by \textit{C. Consani} [Compos. Math. 111, 323-358 (1998; Zbl 0932.14011)].
scientific article; zbMATH DE number 1850799