LEADER 05656nam 22006495 450 001 9910254085903321 005 20200629224955.0 010 $a3-319-27371-X 024 7 $a10.1007/978-3-319-27371-6 035 $a(CKB)3710000000579415 035 $a(EBL)4383696 035 $a(SSID)ssj0001607081 035 $a(PQKBManifestationID)16316136 035 $a(PQKBTitleCode)TC0001607081 035 $a(PQKBWorkID)14897033 035 $a(PQKB)11366623 035 $a(DE-He213)978-3-319-27371-6 035 $a(MiAaPQ)EBC4383696 035 $a(PPN)191705837 035 $a(EXLCZ)993710000000579415 100 $a20160126d2016 u| 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aRigid Geometry of Curves and Their Jacobians /$fby Werner Lütkebohmert 205 $a1st ed. 2016. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2016. 215 $a1 online resource (398 p.) 225 1 $aErgebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics,$x0071-1136 ;$v61 300 $aDescription based upon print version of record. 311 $a3-319-27369-8 320 $aIncludes bibliographical references and index. 327 $aRigid 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 327 $a2.9 Riemann's Vanishing TheoremChapter 3: Formal and Rigid Geometry; 3.1 Canonical Reduction of Af noid Domains; 3.1.1 Functors AK A?K and AKA?K; 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 327 $a4.1 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 327 $a6.5 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 330 $aThis book presents some of the most important aspects of rigid geometry, namely its applications to the study of smooth algebraic curves, of their Jacobians, and of abelian varieties - all of them defined over a complete non-archimedean valued field. The text starts with a survey of the foundation of rigid geometry, and then focuses on a detailed treatment of the applications. In the case of curves with split rational reduction there is a complete analogue to the fascinating theory of Riemann surfaces. In the case of proper smooth group varieties the uniformization and the construction of abelian varieties are treated in detail. Rigid geometry was established by John Tate and was enriched by a formal algebraic approach launched by Michel Raynaud. It has proved as a means to illustrate the geometric ideas behind the abstract methods of formal algebraic geometry as used by Mumford and Faltings. This book should be of great use to students wishing to enter this field, as well as those already working in it. 410 0$aErgebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics,$x0071-1136 ;$v61 606 $aAlgebraic geometry 606 $aFunctions of complex variables 606 $aAlgebraic Geometry$3https://scigraph.springernature.com/ontologies/product-market-codes/M11019 606 $aSeveral Complex Variables and Analytic Spaces$3https://scigraph.springernature.com/ontologies/product-market-codes/M12198 615 0$aAlgebraic geometry. 615 0$aFunctions of complex variables. 615 14$aAlgebraic Geometry. 615 24$aSeveral Complex Variables and Analytic Spaces. 676 $a516.352 700 $aLütkebohmert$b Werner$4aut$4http://id.loc.gov/vocabulary/relators/aut$0349474 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910254085903321 996 $aRigid geometry of curves and their jacobians$91523613 997 $aUNINA