By Werner Lütkebohmert (auth.)

This publication offers the most vital elements of inflexible geometry, particularly its functions to the research of gentle algebraic curves, in their Jacobians, and of abelian kinds - them all outlined over a whole non-archimedean valued box. The textual content starts off with a survey of the root of inflexible geometry, after which specializes in an in depth therapy of the purposes. with regards to curves with break up rational aid there's a whole analogue to the interesting thought of Riemann surfaces. with regards to right soft team forms the uniformization and the development of abelian types are taken care of intimately.

Rigid geometry used to be validated by means of John Tate and was once enriched through a proper algebraic method introduced through Michel Raynaud. It has proved as a method to demonstrate the geometric rules in the back of the summary tools of formal algebraic geometry as utilized by Mumford and Faltings. This ebook can be of serious use to scholars wishing to go into this box, in addition to these already operating in it.

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**Extra resources for Rigid Geometry of Curves and Their Jacobians**

**Example text**

1 (Tate). Let K be a non-Archimedean field of arbitrary characteristic and let q ∈ K × with 0 < |q| < 1. Then the field of meromorphic q-periodic © Springer International Publishing Switzerland 2016 W. Lütkebohmert, Rigid Geometry of Curves and Their Jacobians, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. , F (q) is finitely generated field of transcendence degree 1 over K and of genus 1. More precisely, F (q) = K(℘, ℘), ˜ where n∈Z q nξ − 2 · s1 , (1 − q n ξ )2 n∈Z q 2n ξ 2 + s1 (1 − q n ξ )3 ℘ (ξ ) = ℘(ξ ˜ )= with s := m≥1 m qm 1 − qm for ∈ N.

It suffices to look at a proper algebraic A-scheme Y where A is an affinoid algebra. If Y is projective over A, then we can assume that Y = PAn is already the projective space. For c ∈ |K × | consider the subsets Yi (c) := x ∈ PAn , ξj (x) ≤ c · ξi (x) for j = 0, . . , n for i = 1, . . , n. Then Y (c) := {Y0 (c), . . , Yn (c)} is an admissible affinoid covering of PAn for c ≥ 1 with Yi (1) A Yi (c) for i = 0, . . , n and c > 1. 3. 2] there exists a surjective Amorphism f : Z → Y from a projective A-scheme Z to Y .

In the next section we will study more general group actions than Tate’s action M × P1K → P1K ; (q, z) −→ q · z, on the projective line. The group M is only a special case of a Schottky group; cf. 3, and so Tate’s curves are a special case of Mumford curves; cf. 1. In the following sections we will present much more general results. 1. In this section we will study the structure of those finitely generated subgroups of the projective linear group PGL(2, K), which are free of torsion and act discontinuously on a non-empty open subdomain of the projective line.