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Rigid Geometry of Curves and Their Jacobians

  • The first comprehensive presentation of the whole topic of curves, Jacobians, abelian varieties and proper analytic group varieties over non-archimedian fields

  • Introduces the powerful tools of formal algebraic geometry as used in arithmetic geometry

  • The book builds a bridge to the more advanced research on the moduli of degeneration of abelian varieties which is a central object in arithmetic geometry

Book

Table of contents

  1. Front Matter
    Pages I-XVIII
  2. Werner Lütkebohmert
    Pages 1-28
  3. Werner Lütkebohmert
    Pages 29-101
  4. Werner Lütkebohmert
    Pages 103-175
  5. Werner Lütkebohmert
    Pages 177-216
  6. Werner Lütkebohmert
    Pages 217-254
  7. Werner Lütkebohmert
    Pages 255-308
  8. Werner Lütkebohmert
    Pages 309-353
  9. Back Matter
    Pages 355-386

About this book

Introduction

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.


Keywords

Algebraic Geometry Arithmetic Geometry Number Theory Several Complex Variables and Analytic Spaces Commutative Rings and Algebras

Authors and affiliations

  1. 1.Institute of Pure MathematicsUlm UniversityUlmGermany

Bibliographic information

  • Book Title Rigid Geometry of Curves and Their Jacobians
  • Authors Werner Lütkebohmert
  • Series Title Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
  • Series Abbreviated Title Ergebnisse Mathematik 3.F.
  • DOI https://doi.org/10.1007/978-3-319-27371-6
  • Copyright Information Springer International Publishing Switzerland 2016
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-319-27369-3
  • Softcover ISBN 978-3-319-80123-0
  • eBook ISBN 978-3-319-27371-6
  • Series ISSN 0071-1136
  • Series E-ISSN 2197-5655
  • Edition Number 1
  • Number of Pages XVIII, 386
  • Number of Illustrations 1 b/w illustrations, 0 illustrations in colour
  • Topics Algebraic Geometry
    Several Complex Variables and Analytic Spaces
  • Buy this book on publisher's site
Industry Sectors
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Reviews

“The work is written in a lucid writing style, and details of proofs are always provided. It contains many important contributions to the literature on rigid geometry, both from the point of view of research and from that of exposition. … The book is therefore highly useful both as a standard reference and as a main resource for an advanced graduate course on rigid geometry.” (Jeroen Sijsling, zbMATH 1387.14003, 2018)

“Werner Lütkebohmert presents in his book the rigid analytic analog of classical topics in complex analysis, namely the theory of compact Riemann surfaces and their Jacobian varieties. … It is a comprehensive exposition of this brilliant theory in a single volume and a must-have for everybody learning or knowing rigid analytic geometry.” (Urs Hartl, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 119, 2017)