Three-dimensional hyperbolic geometry as \(\infty\)-adic Arakelov geometry
According to Arakelov's program, to an algebraic variety \(X\) over a number field \(K\) one could attach a completed arithmetical variety \(\overline X\) of dimension \(\dim(X)+1\), \[ \overline X\to\overline{\text{Spec} A}=\text{Spec} A\cup\{\infty_ 1,\ldots,\infty_ r\}, \] where \(A\) is the maximal order, \(\{\infty_ 1,\ldots,\infty_ r\}\) the set of all places at \(\infty\) of \(K\) [see \textit{S. Lang}, ``Introduction to Arakelov theory'' (1988; Zbl 0667.14001)]. In this setting fibers of \(\overline X\) at \(\infty_ j\) are the complex manifolds \(X_{\infty_ j}=X_{\infty_ j}(\mathbb{C})\) endowed with a Hermitian metric. If \(X=X(\mathbb{C})\) is a Riemann surface then the Green function on \(X\) plays the role of local intersection indices ``at the closed fiber at infinity''. In this paper a filling \({\mathcal X}\) of \(X\) is constructed, which is an auxiliary three dimensional manifold endowed with a metric of constant negative curvature, such that \(X\) is its boundary, and which can be defined by Schottky uniformization. It is proven that some Green's functions on \(X\) can be expressed in terms of the geometry of geodesics on \({\mathcal X}\). The author suggests that \({\mathcal X}\), the set of all closed geodesics in \({\mathcal X}\) and the set of geodesics with one end at \(X\) can play the roles, respectively, of `` \(\infty\)-adic completion'' of \(\overline X\), its ``closed fiber at infinity'', and its ``reduction modulo \(\infty\)'', in the spirit of the work of \textit{D. Mumford} [Compos. Math. 24, 129-174 (1972; Zbl 0228.14011)]. Relations with recent work of Ch. Deninger are discussed.
scientific article; zbMATH DE number 12333