Rigid Analytic Geometry and Its Applications, 2004 Progress in Mathematics Series, Vol. 218

Rigid (analytic) spaces were invented to describe degenerations, reductions, and moduli of algebraic curves and abelian varieties. This work, a revised and greatly expanded new English edition of an earlier French text by the same authors, presents important new developments and applications of the theory of rigid analytic spaces to abelian varieties, "points of rigid spaces," étale cohomology, Drinfeld modular curves, and Monsky-Washnitzer cohomology. The exposition is concise, self-contained, rich in examples and exercises, and will serve as an excellent graduate-level text for the classroom or for self-study.

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.

Chapters on the applications of this theory to curves and abelian varieties The work of Drinfeld on "elliptic modules" and the Langlands conjectures for function fields use a background of rigid étale cohomology; detailed treatment of this topic Presentation of the rigid analytic part of Raynaud’s proof of the Abhyankar conjecture for the affine line, with only the rudiments of that theory