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Author Fresnel, Jean. author Rigid Analytic Geometry and Its Applications [electronic resource] / by Jean Fresnel, Marius van der Put Boston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 2004 http://dx.doi.org/10.1007/978-1-4612-0041-3 XI, 299 p. online resource

CONTENT

1 Valued Fields and Normed Spaces -- 1.1 Valued fields -- 1.2 Banach spaces and Banach algebras -- 2 The Projective Line -- 2.1 Some definitions -- 2.2 Holomorphic functions on an affinoid subset -- 2.3 The residue theorem -- 2.4 The Grothendieck topology on P -- 2.5 Some sheaves on P -- 2.6 Analytic subspaces of P -- 2.7 Cohomology on an analytic subspace of P -- 3 Affinoid Algebras -- 3.1 Definition of an affinoid algebra -- 3.2 Consequences of the Weierstrass theorem -- 3.3 Affinoid spaces, Examples -- 3.4 Properties of the spectral (semi-)norm -- 3.5 Integral extensions of affinoid algebras -- 3.6 The differential module ?A/kf -- 3.7 Products of affinoid spaces, Picard groups -- 4 Rigid Spaces -- 4.1 Rational subsets -- 4.2 The weak G-topology and Tateโ{128}{153}s theorem -- 4.3 General rigid spaces -- 4.4 Sheaves on a rigid space -- 4.5 Coherent analytic sheaves -- 4.6 The sheaf of meromorphic functions -- 4.7 Rigid vector bundles -- 4.8 Analytic reductions and formal schemes -- 4.9 Analytic reductions of a subspace of Pk1, an -- 4.10 Separated and proper rigid spaces -- 5 Curves and Their Reductions -- 5.1 The Tate curve -- 5.2 Nรฉron models for abelian varieties -- 5.3 The Nรฉron model of an elliptic curve -- 5.4 Mumford curves and Schottky groups -- 5.5 Stable reduction of curves -- 5.6 A rigid proof of stable reduction for curves -- 5.7 The universal analytic covering of a curve -- 6 Abelian Varieties -- 6.1 The complex case -- 6.2 The non-archimedean case -- 6.3 The analytification of an algebraic torus -- 6.4 Lattices and analytic tori -- 6.5 Meromorphic functions on an analytic torus -- 6.6 Analytic tori and abelian varieties -- 6.7 Nรฉron models and uniformization -- 7 Points of Rigid Spaces, Rigid Cohomology -- 7.1 Points and sheaves on an affinoid space -- 7.2 Explicit examples in dimension 1 -- 7.3 $$\mathcal{P}$$(X) and the reductions of X -- 7.4 Base change for overconvergent sheaves -- 7.5 Overconvergent affinoid spaces -- 7.6 Monsky-Washnitzer cohomology -- 7.7 Rigid cohomology -- 8 Etale Cohomology of Rigid Spaces -- 8.1 Etale morphisms -- 8.2 The รฉtale site -- 8.3 Etale points, overconvergent รฉtale sheaves -- 8.4 Etale cohomology in dimension 1 -- 8.5 Higher dimensional rigid spaces -- 9 Covers of Algebraic Curves -- 9.1 Introducing the problem -- 9.2 I. Serreโ{128}{153}s result -- 9.3 II. Rigid construction of coverings -- 9.4 III. Reductions of curves modulo p -- References -- List of Notation

Mathematics Algebraic geometry Functions of complex variables Geometry Number theory Mathematics Geometry Algebraic Geometry Several Complex Variables and Analytic Spaces Number Theory

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