**University Lecture Series**

Volume: 36;
2005;
136 pp;
Softcover

MSC: Primary 58;

Print ISBN: 978-0-8218-3833-4

Product Code: ULECT/36

List Price: $34.00

Individual Member Price: $27.20

**Electronic ISBN: 978-1-4704-2181-6
Product Code: ULECT/36.E**

List Price: $34.00

Individual Member Price: $27.20

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#### Supplemental Materials

# Arithmetic Noncommutative Geometry

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*Matilde Marcolli*

Arithmetic noncommutative geometry uses ideas and tools from noncommutative geometry to address questions in a new way and to reinterpret results and constructions from number theory and arithmetic algebraic geometry. This general philosophy is applied to the geometry and arithmetic of modular curves and to the fibers at Archimedean places of arithmetic surfaces and varieties.

Noncommutative geometry can be expected to say something about topics of arithmetic interest because it provides the right framework for which the tools of geometry continue to make sense on spaces that are very singular and apparently very far from the world of algebraic varieties. This provides a way of refining the boundary structure of certain classes of spaces that arise in the context of arithmetic geometry.

With a foreword written by Yuri Manin and a brief introduction to noncommutative geometry, this book offers a comprehensive account of the cross fertilization between two important areas, noncommutative geometry and number theory. It is suitable for graduate students and researchers interested in these areas.

#### Table of Contents

# Table of Contents

## Arithmetic Noncommutative Geometry

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents vii8 free
- Foreword ix10 free
- Chapter 1. Ouverture 114 free
- Chapter 2. Noncommutative modular curves 1528
- Chapter 3. Quantum statistical mechanics and Galois theory 5164
- Chapter 4. Noncommutative geometry at arithmetic infinity 8194
- 1. Schottky uniformization 8194
- 2. Dynamics and noncommutative geometry 88101
- 3. Arithmetic infinity: archimedean primes 93106
- 4. Arakelov geometry and hyperbolic geometry 97110
- 5. Intermezzo: quantum gravity and black holes 100113
- 6. Dual graph and noncommutative geometry 105118
- 7. Arithmetic varieties and L-factors 109122
- 8. Archimedean cohomology 115128

- Chapter 5. Vistas 125138
- Bibliography 131144
- Back Cover Back Cover1152

#### Readership

Graduate students and research mathematicians interested in geometry and number theory.

#### Reviews

This book succeeds in giving a beautiful overview of the rapid evolution field of arithmetic noncommutative geometry. It proves, once again, how successful it is to apply techniques developed in one branch of mathematics to problems from another branch, to find out unexpected connections, and reinterpret in a new perspective results and constructions.

-- Zentralblatt MATH

This book is written in a lively style and will be very helpful to young researchers who wish to work in the (growing) area in which number theory and noncommutative geometry interplay.

-- Mathematical Reviews