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Noncommutative Geometry and Number Theory: Where Arithmetic meets Geometry and Physics
Edited by: Caterina Consani, Johns Hopkins University, Baltimore, MD, and Matilde Marcolli, Max-Planck-Institut für Mathematik, Bonn, Germany
A publication of Vieweg+Teubner.
Vieweg Aspects of Mathematics
2006; 372 pp; hardcover
Volume: 37
ISBN-10: 3-8348-0170-4
ISBN-13: 978-3-8348-0170-8
List Price: US$113
Member Price: US$101.70
Order Code: VWAM/37
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In recent years, number theory and arithmetic geometry have been enriched by new techniques from noncommutative geometry, operator algebras, dynamical systems, and \(K\)-Theory. This volume collects and presents up-to-date research topics in arithmetic and noncommutative geometry and ideas from physics that point to possible new connections between the fields of number theory, algebraic geometry and noncommutative geometry. The articles collected in this volume present new noncommutative geometry perspectives on classical topics of number theory and arithmetic such as modular forms, class field theory, the theory of reductive \(p\)-adic groups, Shimura varieties, the local \(L\)-factors of arithmetic varieties. They also show how arithmetic appears naturally in noncommutative geometry and in physics, in the residues of Feynman graphs, in the properties of noncommutative tori, and in the quantum Hall effect.

A publication of Vieweg+Teubner. The AMS is exclusive distributor in North America. Vieweg+Teubner Publications are available worldwide from the AMS outside of Germany, Switzerland, Austria, and Japan.


Graduate students and research mathematicians interested in geometry and topology.

Table of Contents

  • A.-M. Aubert, P. Baum, and R. Plymen -- The Hecke algebra of a reductive \(p\)-adic group: a view from noncommutative geometry
  • D. Blasius -- Hilbert modular forms and the Ramanujan conjecture
  • F. P. Boca and A. Zaharescu -- Farey fractions and two-dimensional tori
  • A. Connes and H. Moscovici -- Transgression of the Godbillon-Vey class and Rademacher functions
  • C. Consani and M. Marcolli -- Archimedean cohomology revisited
  • A. Fel'shtyn and E. Troitsky -- A twisted Burnside theorem for countable groups and Reidemeister numbers
  • M. Khalkhali and B. Rangipour -- Introduction to Hopf cyclic cohomology
  • M. Kim -- The non-abelian (or non-linear) method of Chabauty
  • D. Kreimer -- The residues of quantum field theory--numbers we should know
  • M. Laca and M. van Frankenhuijsen -- Phase transitions with spontaneous symmetry breaking on Hecke \(C^\ast\)-algebras from number fields
  • G. Landi -- On harmonic maps in noncommutative geometry
  • M. Marcolli and V. Mathai -- Towards the fractional quantum Hall effect: a noncommutative geometry perspective
  • R. Meyer -- Homological algebra for Schwartz algebras of reductive \(p\)-adic groups
  • V. Nistor -- A non-commutative geometry approach to the representation theory of reductive \(p\)-adic groups: Homology of Hecke algebras, a survey and some new results
  • F. Paugam -- Three examples of non-commutative boundaries of Shimura varieties
  • A. Polishchuk -- Holomorphic bundles on 2-dimensional noncommutative toric orbifolds
  • R. Ponge -- A new short proof of the local index formula of Atiyah-Singer
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