The theory of rigid (analytic) spaces, originally invented to describe degenerations, reductions, and moduli of algebraic curves and abelian varieties, has undergone significant growth in the last two decades; today the theory has applications to arithmetic algebraic geometry, number theory, the arithmetic of function fields, and p-adic differential equations. This work, a revised and greatly expanded new English edition of the earlier French text by the same authors, is an accessible introduction to the theory of rigid spaces and now includes a large number of exercises.
Key topics:
- Chapters on the applications of this theory to curves and abelian varieties: the Tate curve, stable reduction for curves, Mumford curves, N?ron models, uniformization of abelian varieties
- Unified treatment of the concepts: points of a rigid space, overconvergent sheaves, Monsky--Washnitzer cohomology and rigid cohomology; detailed examination of Kedlaya's application of the Monsky--Washnitzer cohomology to counting points on a hyperelliptic curve over a finite field
- The work of Drinfeld on "elliptic modules" and the Langlands conjectures for function fields use a background of rigid ?tale 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
A basic knowledge of algebraic geometry is a sufficient prerequisite for this text. Advanced graduate students and researchers in algebraic geometry, number theory, representation theory, and other areas of mathematics will benefit from the book's breadth and clarity. TOC:Preface.- Valued fields and normed spaces.- The projective line.- Affinoid algebras.- Rigid spaces.- Curves and their reductions.- Abelian varieties.- Points of rigid spaces, rigid cohomology.- Etale cohomology of rigid spaces.- Covers of algebraic curves.- References.- List of Notation.- Index.
