Berkovich Curves and Schottky Uniformization II: Analytic Uniformization of Mumford Curves

Schottky spaces and universal Mumford curves over $${\mathbb {Z}}$$

“…We would also like to refer the reader to the recent text [PT21] by the authors, which provides a detailed exposition of the theory non-archimedean Schottky groups and Schottky uniformization in the setting of Berkovich spaces and fixes some gaps in the existing literature. Our treatment here follows this reference quite closely.…”

“…Given a non-archimedean field k, the group PGL 2 (k) acts naturally on the Berkovich projective line P 1,an k and one can describe Schottky uniformization of Mumford curves in complete analogy with the complex case. A study of classical results on Schottky groups and Mumford curves in this new framework was initiated in Berkovich's first monograph (see [Ber90,§4.4]) and has been expanded in recent work by the authors [PT21], whose appendix contains a description of striking applications of these results.…”

“…In Section 3, we formulate classical definitions and results of the theory of Schottky groups in a way that applies to any valued field, archimedean and non-archimedean alike. We adapt classical notions in the non-archimedean theory to the framework of Berkovich spaces, relying on previous work [PT21] by the authors. In Section 4, we define the space S g as a Berkovich space over Z and we prove that it is open in A 3g−3,an Z .…”

“…This allows to define a continuous function Mumf g,a → M trop g sending (the class of) a Mumford curve in (the class of) its skeleton. For a point x ∈ Mumf g,a , recall that the Schottky uniformization P 1,an H(x) − L x → C x restricts to a universal cover of the skeleton Σ x (see [PT21,Theorem 6.4.18]). Via this restriction, the Schottky group Γ x ∼ = π 1 (C x ) can be identified with the topological fundamental group π 1 (Σ x ).…”