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129 pages, 2003

Abstract:

This book contains a well-organized "overview of recent results illustrating the interplay between noncommutative geometry and arithmetic geometry/number theory", as well as results in noncommutative geometry (NCG) at arithmetic infinity and emphasis on NCG applications to quantum theories. Table of Contents: "Chapter 1. Ouverture p.5 1. The NCG dictionary 7 2. Noncommutative spaces 8 3. Spectral triples 9 Chapter 2. Noncommutative modular curves 17 1. Modular curves 17 2. The noncommutative boundary of modular curves 24 3. Modular interpretation: noncommutative elliptic curves 24 4. Limiting modular symbols 29 5. Hecke eigenforms 40 6. Selberg zeta function 43 7. The modular complex and K-theory of C*--algebras 44 8. Intermezzo: Chaotic Cosmology 45 Chapter 3. Quantum statistical mechanics and Galois theory 53 1. Quantum Statistical Mechanics 54 2. The Bost--Connes system 58 3. Noncommutative Geometry and Hilbert's 12th problem 63 4. The GL2 system 65 5. Quadratic fields 72 Chapter 4. Noncommutative geometry at arithmetic infinity 81 1. Schottky uniformization 81 2. Dynamics and noncommutative geometry 88 3. Arithmetic infinity: archimedean primes 93 4. Arakelov geometry and hyperbolic geometry 96 5. Intermezzo: Quantum gravity and black holes 100 6. Dual graph and noncommutative geometry 104 7. Arithmetic varieties and L--factors 109 8. Archimedean cohomology 115 Bibliography p.125" http://xxx.lanl.gov/PS_cache/math/pdf/0409/0409520v1.pdf

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Open access: http://xxx.lanl.gov/PS_cache/math/pdf/0409/0409520v1.pdf

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