Rigid geometry of curves and their Jacobians
This book presents some of the most important aspects of rigid geometry, namely its applications to the study of smooth algebraic curves, of their Jacobians, and of abelian varieties - all of them defined over a complete non-archimedean valued field. The text starts with a survey of the foundation of rigid geometry, and then focuses on a detailed treatment of the applications. In the case of curves with split rational reduction there is a complete analogue to the fascinating theory of Riemann s…
Read More ↓Title Statements
- Full title :Rigid geometry of curves and their Jacobians / Werner Lütkebohmert.
Summary
This book presents some of the most important aspects of rigid geometry, namely its applications to the study of smooth algebraic curves, of their Jacobians, and of abelian varieties - all of them defined over a complete non-archimedean valued field. The text starts with a survey of the foundation of rigid geometry, and then focuses on a detailed treatment of the applications. In the case of curves with split rational reduction there is a complete analogue to the fascinating theory of Riemann surfaces. In the case of proper smooth group varieties the uniformization and the construction of abelian varieties are treated in detail. Rigid geometry was established by John Tate and was enriched by a formal algebraic approach launched by Michel Raynaud. It has proved as a means to illustrate the geometric ideas behind the abstract methods of formal algebraic geometry as used by Mumford and Faltings. This book should be of great use to students wishing to enter this field, as well as those already working in it.
Read More ↓Edition Statements
- Publication Date:2016
Publication Statements
- Publisher(s):Springer
- Place of Publication:Cham
- Language:ger
- Publication Date:2016
The Physical Object
- Description:1 online resource
Edition Identifiers
- ISBN:9783319273716, 331927371X
- DOI:10.1007/978-3-319-27371-6
- oclc:ocn936176794
- LC Classification:QA567
- Dewey Decimal:516.3/52
Table of Contents
Rigid Geometry of Curves and Their Jacobians; Introduction; Chapter 1: Classical Rigid Geometry; 1.1 Non-Archimedean Fields; 1.2 Restricted Power Series; 1.3 Af noid Spaces; 1.4 The Maximum Principle; 1.5 Rigid Analytic Spaces; 1.6 Coherent Sheaves; 1.7 Line Bundles; 1.8 Algebraization of Proper Rigid Analytic Curves; Chapter 2: Mumford Curves; 2.1 Tate's Elliptic Curve; 2.2 Schottky Groups; 2.3 De nition and Properties; 2.4 Skeletons; 2.5 Automorphic Functions; 2.6 Drinfeld's Polarization; 2.7 Rigid Analytic Tori and Their Duals; 2.8 Jacobian Variety of a Mumford Curve.
2.9 Riemann's Vanishing TheoremChapter 3: Formal and Rigid Geometry; 3.1 Canonical Reduction of Af noid Domains; 3.1.1 Functors AK ÅK and AKÃK; 3.1.2 Formal Analytic Spaces; 3.1.3 Finiteness Theorem of Grauert-Remmert-Gruson; 3.2 Admissible Formal Schemes; 3.3 Generic Fiber of Admissible Formal Schemes; 3.4 Reduced Fiber Theorem; 3.4.1 Analytic Method of Grauert-Remmert-Gruson; 3.4.2 Elementary Method of Epp; 3.4.3 The Natural Approach; 3.5 Complements on Flatness; 3.6 Approximation in Smooth Rigid Spaces; 3.7 Compacti cation of Smooth Curve Fibrations; Chapter 4: Rigid Analytic Curves.
4.1 Formal Fibers4.2 Genu…
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