Rigid Analytic Geometry and Its Applications

✍ Scribed by Jean Fresnel, Marius van der Put

• Publisher: Birkhäuser Boston
• Year: 2003
• Tongue: English
• Leaves: 306
• Series: Progress in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

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," ?tale 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.

✦ Table of Contents

Contents......Page page000004.djvu
1.1. Valued fields......Page page000012.djvu
1.2. Banach spaces and Banach algebras......Page page000015.djvu
2.1. Some definitions......Page page000024.djvu
2.2. Holomorphic functions on an affinoid subset......Page page000026.djvu
2.3. The residue theorem......Page page000033.djvu
2.4. The Grothendieck topology on P......Page page000036.djvu
2.5. Some sheaves on P......Page page000041.djvu
2.6. Analytic subspaces of P......Page page000044.djvu
2.7. Cohomology on an analytic subspace of P......Page page000048.djvu
3. Affinoid Algebras......Page page000056.djvu
3.1. Definition of an affinoid algebra......Page page000057.djvu
3.2. Consequences of the Weierstrass theorem......Page page000059.djvu
3.3. Affinoid spaces, Examples......Page page000062.djvu
3.4. Properties of the spectral (semi-)norm......Page page000066.djvu
3.5. Integral extensions of affinoid algebras......Page page000070.djvu
3.6. The differential module Ωᶠ_A/k......Page page000074.djvu
3.7. Products of affinoid spaces, Picard groups......Page page000077.djvu
4.2. The weak G-topology and Tate's theorem......Page page000082.djvu
4.3. General rigid spaces......Page page000092.djvu
4.4. Sheaves on a rigid space......Page page000097.djvu
4.5. Coherent analytic sheaves......Page page000098.djvu
4.6. The sheaf of meromorphic functions......Page page000102.djvu
4.7. Rigid vector bundles......Page page000108.djvu
4.8. Analytic reductions and formal schemes......Page page000110.djvu
4.9. Analytic reductions of a subspace of P^1,an_k......Page page000118.djvu
4.10. Separated and proper rigid spaces......Page page000121.djvu
5.1. The Tate curve......Page page000132.djvu
5.2. Néron models for abelian varieties......Page page000144.djvu
5.3. The Néron model of an elliptic curve......Page page000147.djvu
5.4. Mumford curves and Schottky groups......Page page000151.djvu
5.5. Stable reduction of curves......Page page000153.djvu
5.6. A rigid proof of stable reduction for curves......Page page000158.djvu
5.7. The universal analytic covering of a curve......Page page000171.djvu
6.1. The complex case......Page page000176.djvu
6.3. The analytification of an algebraic torus......Page page000178.djvu
6.4. Lattices and analytic tori......Page page000181.djvu
6.5. Meromorphic functions on an analytic torus......Page page000183.djvu
6.6. Analytic tori and abelian varieties......Page page000185.djvu
6.7.1. Néron model for an abelian variety of the form T......Page page000188.djvu
6.7.2. Rigid construction of more general abelian varieties......Page page000191.djvu
6.7.3. Stable reduction for abelian varieties......Page page000195.djvu
6.7.4. Uniformization of a Jacobian variety......Page page000196.djvu
7. Points of Rigid Spaces, Rigid Cohomology......Page page000202.djvu
7.1. Points and sheaves on an affinoid space......Page page000203.djvu
7.2. Explicit examples in dimension 1......Page page000214.djvu
7.3. P(X) and the reductions of X......Page page000217.djvu
7.4. Base change for overconvergent sheaves......Page page000219.djvu
7.5. Overconvergent affinoid spaces......Page page000225.djvu
7.6. Monsky-Washnitzer cohomology......Page page000235.djvu
7.7. Rigid cohomology......Page page000247.djvu
8. Etale Cohomology of Rigid Spaces......Page page000250.djvu
8.1. Etale morphisms......Page page000251.djvu
8.2. The etale site......Page page000255.djvu
8.3. Etale points, overconvergent étale sheaves......Page page000259.djvu
8.4. Etale cohomology in dimension 1......Page page000263.djvu
8.5. Higher dimensional rigid spaces......Page page000266.djvu
9.1. Introducing the problem......Page page000270.djvu
9.2. I. Serre's result......Page page000274.djvu
9.3. II. Rigid construction of coverings......Page page000275.djvu
9.3.1. GAGA for coherent sheaves on P......Page page000276.djvu
9.3.2. Covers......Page page000278.djvu
9.3.3. Ramified coverings of P¹_K......Page page000280.djvu
9.4. III. Reductions of curves modulo p......Page page000283.djvu
References......Page page000286.djvu
List of Notation......Page page000300.djvu
Index......Page page000304.djvu