**Mathematical Surveys and Monographs**

Volume: 33;
1990;
169 pp;
Softcover

MSC: Primary 12; 32; 14; 20; 46;
Secondary 47

**Print ISBN: 978-0-8218-9020-2
Product Code: SURV/33.S**

List Price: $76.00

AMS Member Price: $60.80

MAA Member Price: $68.40

**Electronic ISBN: 978-1-4704-1260-9
Product Code: SURV/33.S.E**

List Price: $71.00

AMS Member Price: $56.80

MAA Member Price: $63.90

# Spectral Theory and Analytic Geometry over Non-Archimedean Fields

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*Vladimir G. Berkovich*

The purpose of this book is to introduce a new notion of analytic space over a non-Archimedean field. Despite the total disconnectedness of the ground field, these analytic spaces have the usual topological properties of a complex analytic space, such as local compactness and local arcwise connectedness. This makes it possible to apply the usual notions of homotopy and singular homology. The book includes a homotopic characterization of the analytic spaces associated with certain classes of algebraic varieties and an interpretation of Bruhat-Tits buildings in terms of these analytic spaces. The author also studies the connection with the earlier notion of a rigid analytic space. Geometrical considerations are used to obtain some applications, and the analytic spaces are used to construct the foundations of a non-Archimedean spectral theory of bounded linear operators. This book requires a background at the level of basic graduate courses in algebra and topology, as well as some familiarity with algebraic geometry. It would be of interest to research mathematicians and graduate students working in algebraic geometry, number theory, and \(p\)-adic analysis.

#### Table of Contents

# Table of Contents

## Spectral Theory and Analytic Geometry over Non-Archimedean Fields

- Contents vii8
- Introduction 112
- Chapter 1. The Spectrum of a Commutative Banach Ring 1122
- Chapter 2. Affinoid Spaces 2132
- Chapter 3. Analytic Spaces 4758
- Chapter 4. Analytic Curves 7586
- Chapter 5. Analytic Groups and Buildings 93104
- 5.1. Subgroups and homogeneous spaces of analytic groups 93104
- 5.2. Peaked points and the *-multiplication 97108
- 5.3. A class of affinoid subgroups of a Chevalley group 103114
- 5.4. Two embeddings of the building B{G) in G[sup(an)] 108119
- 5.5. Embeddings of the building B(G) into compacthomogeneous spaces (G/H)[sup(an)] 111122

- Chapter 6. The Homotopy Type of Certain Analytic Spaces 115126
- 6.1. Projective space and its subsets 115126
- 6.2. Chevalley groups and their compact homogeneous spaces 118129
- 6.3. Analytic tori 119130
- 6.4. Analytic groups with potentially good reduction 121132
- 6.5. Abelian varieties 124135
- 6.6. Weak local contractibility of the two-dimensional closed disc 125136

- Chapter 7. Spectral Theory 127138
- Chapter 8. Perturbation Theory 139150
- Chapter 9. The Dimension of a Banach Algebra 153164
- References 161172
- Index 165176