nLab arithmetic and noncommutative geometry

A relation between noncommutative geometry and arithmetic (= number theory) has been explored much in the work of Alain Connes and his collaborators, especially Marcolli and Consani.

Some surveys include

• P. Almeida, Noncommutative geometry and arithmetics, Russian Journal of Mathematical Physics 16, No. 3, 2009, pp. 350–362, doi
• Matilde Marcolli, Lectures on arithmetic noncommutative geometry, math/0409520

There is also another line of thought in the work Manin-Marcolli on the relation of Arakelov geometry and noncommutative geometry. Arakelov’s geometry is of course, motivated by number theory.

Independetly, one should also notice that the noncommutative geometry over non-archimedean fields is relevant for homological mirror symmetry as explored in the works of Maxim Kontsevich and Yan Soibelman.