Arakelov Theory

Abstract

In 1974 Arakelov indicated a way to complete a family of curves over the ring of integers of a number field by including the fibers at infinity. This amounted to the corresponding Riemann surfaces and their differential geometric properties once the number field gets imbedded into the complex numbers. Arakelov showed how one could define a global intersection number for two arithmetic curves on an arithmetic surface, and that this intersection number was actually defined on the rational equivalence classes, thus providing the beginning for the ultimate transposition of all algebraic geometry to this case. This is a huge program, which combines the algebraic side of algebraic geometry, the complex analytic side, complex differential geometry, partial differential equations and Laplace operators with needed estimates on the eigenvalues, in a completely open ended unification of mathematics as far as one can see.