The main focus of these lecture notes is an interpretation of the arithmetic of modular curves and archimedean fibers of arithmetic varieties in the realm of non-commutative geometry. The first chapter reviews the basic notions of non-commutative geometry. The second chapter discusses properties of modular curves from the point of view of a description of their boundary strata in the framework of non-commutative geometry. The third chapter includes a discussion of the non-commutative space of commensurability classes of Q-lattices and its relation to class field theory. The fourth and last chapter involves the non-commutative geometry of the fibers at arithmetic infinity of varieties (surfaces) over number fields. The book aims to introduce and link together a variety of general concepts from geometry, number theory and physics. Many proofs have not been included in the text.

Reviewer:

pso