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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.
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Résumé
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.
Texte suppl.
“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)
Commentaire
"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)
