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 fields has now reached an important 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 connections between the fields of number theory, algebraic geometry and noncommutative geometry.
The contributions to this volume partly reflect the two workshops "Noncommutative Geometry and Number Theory" that took place at the Max-Planck-Institut für Mathematik in Bonn, in August 2003 and June 2004. The two workshops were the first activity entirely dedicated to the interplay between these two fields of mathematics. An important part of the activities, which is also reflected in this volume, came from the hindsight of physics which often provides new perspectives on number theoretic problems that make it possible to employ the tools of noncommutative geometry, well designed to describe the quantum world.
Some contributions to the volume (Aubert-Baum-Plymen, Meyer, Nistor) center on the theory of reductive p-adic groups and their Hecke algebras, a promising direction where noncommutative geometry provides valuable tools for the study of objects of number theoretic and arithmetic interest. A generalization of the classical Burnside theorem using noncommutative geometry is discussed in the paper by Fel'shtyn and Troitsky. The contribution of Laca and van Frankenhuijsen represents another direction in which substantial progress was recently made in applying tools of noncommutative geometry to number theory: the construction of quantum statistical mechanical systems associated to number fields and the relation of their KMS equilibrium states to abelian class field theory. The theory of Shimura varieties is considered from the number theoretic side in the contribution of Blasius, on the Weight-Monodromy conjecture for Shimura varieties associated to quaternion algebras over totally real fields and the Ramanujan conjecture for Hubert modular forms. An approach via noncommutative geometry to the boundaries of Shimura varieties is discussed in Paugam's paper. Modular forms can be studied using techniques from noncommutative geometry, via the Hopf algebra symmetries of the modular Hecke algebras, as discussed in the paper of Connes and Moscovici. The general underlying theory of Hopf cyclic cohomology in noncommutative geometry is presented in the paper of Khalkhali and Rangipour. Further results in arithmetic geometry include a reinterpretation of the archimedean cohomology of the fibers at archimedean primes of arithmetic varieties (Consani-Marcolli), and Kim's paper on a noncommutative method of Chabauty. Arithmetic aspects of noncommutative tori are also discussed (Boca-Zaharescu and Polishchuk). The input from physics and its interactions with number theory and noncommutative geometry is represented by the contributions of Kreimer, Landi, Marcolli-Mathai, and Ponge.
The workshops were generously funded by the Humboldt Foundation and the ZIP Program of the German Federal Government, through the Sofja Kovalevskaya Prize awarded to Marcolli in 2001.
We are very grateful to Yuri Manin and to Alain Connes, who helped us with the organization of the workshops, and whose ideas and results contributed crucially to bring number theory in touch with noncommutative geometry.
Caterina Consani and Matilde Marcolli