Arithmetic Noncommutative Geometry (University Lecture Series)

Arithmetic noncommutative geometry uses ideas and tools from noncommutative geometry to address questions in a new way and to reinterpret results and constructions from number theory and arithmetic algebraic geometry. This general philosophy is applied to the geometry and arithmetic of modular curves and to the fibers at Archimedean places of arithmetic surfaces and varieties. Noncommutative geometry can be expected to say something about topics of arithmetic interest because it provides the right framework for which the tools of geometry continue to make sense on spaces that are very singular and apparently very far from the world of algebraic varieties. This provides a way of refining the boundary structure of certain classes of spaces that arise in the context of arithmetic geometry. With a foreword written by Yuri Manin and a brief introduction to noncommutative geometry, this book offers a comprehensive account of the cross fertilization between two important areas, noncommutative geometry and number theory. It is suitable for graduate students and researchers interested in these areas.

This book succeeds in giving a beautiful overview of the rapid evolution field of arithmetic noncommutative geometry. It proves, once again, how successful it is to apply techniques developed in one branch of mathematics to problems from another branch, to find out unexpected connections, and reinterpret in a new perspective results and constructions. --Zentralblatt MATH

This book is written in a lively style and will be very helpful to young researchers who wish to work in the (growing) area in which number theory and noncommutative geometry interplay. --Mathematical Reviews