Noncommutative Geometry and Number Theory

1. Front Matter

   Pages i-viii

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2. The Hecke algebra of a reductive p-adic group: a geometric conjecture

   • Anne-Marie Aubert, Paul Baum, Roger Plymen

   Pages 1-34

3. Hilbert modular forms and the Ramanujan conjecture

   • Don Blasius

   Pages 35-56

4. Farey fractions and two-dimensional tori

   • Florin P. Boca, Alexandru Zaharescu

   Pages 57-77

5. Transgressions of the Godbillon-Vey Class and Rademacher functions

   • Alain Connes, Henri Moscovici

   Pages 79-107

6. Archimedean cohomology revisited

   • Caterina Consani, Matilde Marcolli

   Pages 109-140

7. A twisted Burnside theorem for countable groups and Reidemeister numbers

   • Alexander Fel’shtyn, Evgenij Troitsky

   Pages 141-154

8. Introduction to Hopf-Cyclic Cohomology

   • Masoud Khalkhali, Bahram Rangipour

   Pages 155-178

9. The non-abelian (or non-linear) method of Chabauty

   • Minhyong Kim

   Pages 179-185

10. The residues of quantum field theory - numbers we should know

    • Dirk Kreimer

    Pages 187-203

11. Phase transitions with spontaneous symmetry breaking on Hecke C*-algebras from number fields

    • Marcelo Laca, Machiel van Frankenhuijsen

    Pages 205-216

12. On harmonic maps in noncommutative geometry

    • Giovanni Landi

    Pages 217-234

13. Towards the fractional quantum Hall effect: a noncommutative geometry perspective

    • Matilde Marcolli, Varghese Mathai

    Pages 235-261

14. Homological algebra for Schwartz algebras of reductive p-adic groups

    • Ralf Meyer

    Pages 263-300

15. A non-commutative geometry approach to the representation theory of reductive p-adic groups: Homology of Hecke algebras, a survey and some new results

    • Victor Nistor

    Pages 301-321

16. Three examples of non-commutative boundaries of Shimura varieties

    • Frédéric Paugam

    Pages 323-340

17. Holomorphic bundles on 2-dimensional noncommutative toric orbifolds

    • A. Polishchuk

    Pages 341-359

18. A New short proof of the local index formula of Atiyah-Singer

    • Raphaël Ponge

    Pages 361-372

In recent years, number theory and arithmetic geometry have been enriched by new techniques from noncommutative geometry, operator algebras, dynamical systems, and K-Theory. Research across these ?elds has now reached an imp- tant turning point, as shows the increasing interest with which the mathematical community approaches these topics. This volume collects and presents up-to-date research topics in arithmetic and noncommutative geometry and ideas from physics that point to possible new c- nections between the ?elds of number theory, algebraic geometry and noncom- tative geometry. Thecontributionstothisvolumepartlyre?ectthetwoworkshops“Noncom- tative Geometry and Number Theory” that took place at the Max–Planck–Institut f¨ ur Mathematik in Bonn, in August 2003 and June 2004. The two workshops were the ?rst activity entirely dedicated to the interplay between these two ?elds of mathematics. An important part of the activities, which is also re?ected in this volume, came from the hindsight of physics which often provides new perspectives onnumber theoretic problems that make it possible to employ the tools of nonc- mutative geometry, well designed to describe the quantum world.

• Archimedean cohomology
• Farey fractions
• Formula of Atiyah-Singer
• Fractional quantum Hall effect
• Hecke algebra
• Hopf cyclic cohomology
• Method of Chabauty
• Rademacher functions
• Ramanujan conjecture
• Reidemeister numbers
• Schwartz algebra
• number theory

• Department of Mathematics, The Johns Hopkins University, Baltimore, USA

  Caterina Consani

• Max-Planck-Institut für Mathematik, Bonn

  Matilde Marcolli

Prof. Dr. Caterina Consani, Department of Mathematics, The Johns Hopkins University, Baltimore, USA
Prof. Dr. Matilde Marcolli, Max-Planck Institute for Mathematics, Bonn, Germany

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