Rigid Geometry of Curves and Their Jacobians / by Werner Lütkebohmert.

Format:

Book

Author/Creator:

Lütkebohmert, Werner., Author.

Series:

Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 0071-1136 ; 61

Language:

English

Subjects (All):

Geometry, Algebraic.

Functions of complex variables.

Algebraic Geometry.

Several Complex Variables and Analytic Spaces.

Local Subjects:

Algebraic Geometry.

Several Complex Variables and Analytic Spaces.

Physical Description:

1 online resource (398 p.)

Edition:

1st ed. 2016.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2016.

Language Note:

English

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.

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 Genus Formula; 4.3 Meromorphic Functions; 4.4 Formal Stable Reduction; 4.5 Stable Reduction; 4.6 Universal Covering of a Curve; 4.7 Characterization of Mumford Curves; Chapter 5: Jacobian Varieties; 5.1 Jacobian of a Smooth Projective Curve; 5.2 Generalized Jacobian of a Semi-Stable Curve; 5.3 Lifting of the Jacobian of the Reduction; 5.4 Morphisms to Rigid Analytic Groups with Semi-Abelian Reduction; 5.5 Uniformization of Jacobians; 5.6 Applications to Abelian Varieties; Chapter 6: Raynaud Extensions; 6.1 Basic Facts; 6.2 Line Bundles; 6.3 Duality; 6.4 Algebraization

6.5 Polarization of Jacobians6.6 Parameterizing Degenerating Abelian Varieties; Chapter 7: Abeloid Varieties; 7.1 Basic Facts on Abeloid Varieties; 7.2 Generation of Subgroups by Smooth Covers; 7.3 Extension of Formal Tori; 7.4 Morphisms from Curves to Groups; 7.5 Stable Reduction of Relative Curves; 7.6 The Structure Theorem; 7.7 Proof of the Structure Theorem; Appendix: Miscellaneous; A.1 Some Notions about Graphs; A.2 Torus Extensions of Formal Abelian Schemes; A.3 Cubical Structures; Glossary of Notations; References; Index

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

ISBN:

3-319-27371-X