Arakelov theory

Summary

Arakelov theory is an intersection theory for arithmetic varieties. These are schemes that are flat and of finite type over the ring of integers of a number field. Such a scheme behaves like a family over a curve that is missing its points at infinity, which correspond to the archimedean places. Arakelov added these fibers at infinity by equipping the associated complex manifolds with Hermitian metrics and Green functions, so that intersection numbers become well defined. The theory gives a geometric meaning to heights and is a main tool of Diophantine geometry.