Noncommutative Geometry and Number Theory: Where Arithmetic meets Geometry and Physics

✍ Scribed by Anne-Marie Aubert, Paul Baum, Roger Plymen (auth.), Prof. Dr. Caterina Consani, Prof. Dr. Matilde Marcolli (eds.)

• Publisher: Vieweg+Teubner Verlag
• Year: 2006
• Tongue: English
• Leaves: 373
• Series: Aspects of Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

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 Lfactors 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.

✦ Table of Contents

Front Matter....Pages i-viii
The Hecke algebra of a reductive p -adic group: a geometric conjecture....Pages 1-34
Hilbert modular forms and the Ramanujan conjecture....Pages 35-56
Farey fractions and two-dimensional tori....Pages 57-77
Transgressions of the Godbillon-Vey Class and Rademacher functions....Pages 79-107
Archimedean cohomology revisited....Pages 109-140
A twisted Burnside theorem for countable groups and Reidemeister numbers....Pages 141-154
Introduction to Hopf-Cyclic Cohomology....Pages 155-178
The non-abelian (or non-linear) method of Chabauty....Pages 179-185
The residues of quantum field theory - numbers we should know....Pages 187-203
Phase transitions with spontaneous symmetry breaking on Hecke C * -algebras from number fields....Pages 205-216
On harmonic maps in noncommutative geometry....Pages 217-234
Towards the fractional quantum Hall effect: a noncommutative geometry perspective....Pages 235-261
Homological algebra for Schwartz algebras of reductive p-adic groups....Pages 263-300
A non-commutative geometry approach to the representation theory of reductive p -adic groups: Homology of Hecke algebras, a survey and some new results....Pages 301-321
Three examples of non-commutative boundaries of Shimura varieties....Pages 323-340
Holomorphic bundles on 2-dimensional noncommutative toric orbifolds....Pages 341-359
A New short proof of the local index formula of Atiyah-Singer....Pages 361-372

✦ Subjects

Mathematics, general