Rigid analytic geometry and its applications

Series: Progress in Mathematics

Jean Fresnel, Marius Van der Put

308 pages, parution le 09/12/2003

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Résumé

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, Néron 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 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

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.

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

L'auteur - Jean Fresnel

Jean Fresnel et Michel Matignon sont professeurs émérites de mathématiques à l'Université Bordeaux-1. Tous deux ont une longue expérience de la préparation des candidats au concours de l'Agrégation de mathématiques. Leurs recherches portent essentiellement sur la théorie des nombres et la géométrie algébrique.

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L'auteur - Marius Van der Put

University of Groningen, The Netherlands

Caractéristiques techniques

	PAPIER
Éditeur(s)	Birkhäuser
Auteur(s)	Jean Fresnel, Marius Van der Put
Parution	09/12/2003
Nb. de pages	308
Format	16 x 24
Couverture	Relié
Poids	570g
Intérieur	Noir et Blanc
EAN13	9780817642068
ISBN13	978-0-8176-4206-8