Contents
Foreword i x
Chapter 1. Ouvertur e 1
1. Th e NC G dictionar y 3
2. Noncommutativ e space s 4
3. Spectra l triple s 6
4. Wh y noncommutativ e geometry ? 12
Chapter 2 . Noncommutativ e modula r curve s 15
1. Modula r curve s 15
2. Th e noncommutativ e boundar y o f modular curve s 2 2
3. Limitin g modula r symbol s 2 7
4. Heck e eigenforms 3 8
5. Selber g zet a functio n 4 1
6. Th e modula r comple x an d if-theor y o f C*-algebra s 4 2
7. Intermezzo : chaoti c cosmolog y 4 4
Chapter 3 . Quantu m statistica l mechanic s an d Galoi s theory 5 1
1. Quantu m statistica l mechanic s 5 3
2. Th e Bost-Conne s syste m 5 6
3. Noncommutativ e geometr y an d Hilbert' s 12th proble m 6 1
4. Th e GL
2
syste m 6 4
5. Quadrati c fields 7 0
Chapter 4 . Noncommutativ e geometr y a t arithmeti c infinit y 8 1
1. Schottk y uniformizatio n 8 1
2. Dynamic s an d noncommutativ e geometr y 8 8
3. Arithmeti c infinity : archimedea n prime s 9 3
4. Arakelo v geometr y an d hyperboli c geometr y 9 7
5. Intermezzo : quantu m gravit y an d blac k hole s 100
6. Dua l grap h an d noncommutativ e geometr y 105
7. Arithmeti c varietie s an d L- factors 109
8. Archimedea n cohomolog y 115
Chapter 5 . Vista s 125
Bibliography 13
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