Rigid Analytic Geometry and Its Applications

Rigid Analytic Geometry and Its Applications by Jean Fresnel, Marius van der Put.

Saved in:
Bibliographic Details
Main Authors: Fresnel, Jean (Author), van der Put, Marius (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: Boston, MA : Birkhäuser Boston : Imprint: Birkhäuser, 2004.
Edition:1st ed. 2004.
Series:Progress in Mathematics, 218
Springer eBook Collection.
Subjects:
Geometry.
Algebraic geometry.
Functions of complex variables.
Number theory.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • 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 Tate’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 Néron models for abelian varieties
  • 5.3 The Néron 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 Néron 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 étale site
  • 8.3 Etale points, overconvergent étale 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. Serre’s result
  • 9.3 II. Rigid construction of coverings
  • 9.4 III. Reductions of curves modulo p
  • References
  • List of Notation.

Similar Items