Arithmetic Noncommutative Geometry

Arithmetic noncommutative geometry denotes the use of ideas and tools from the field of noncommutative geometry, to address questions and reinterpret in a new perspective 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. The main reason why noncommutative geometry can be expected to say something about topics of arithmetic interest lies in the fact that it provides the right framework in 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, such as moduli spaces (of which modular curves are the simplest case) or arithmetic varieties (completed by suitable 'fibers at infinity'), by adding boundaries that are invisible to algebraic geometry, such as degenerations of elliptic curves to noncommutative tori. The text of the book is organized around series of invited lectures delivered by the author at various universities, and the results presented are based on work of the author in collaboration with Alain Connes, Katia Consani, Yuri Manin, and Niranjan Ramachandran.