Affine and mod-affine varieties in arithmetic geometry

François Charles

We will explain how studying arithmetic versions of affine schemes and their birational modifications leads to a generalization to arbitrary schemes of both Fekete's theorem on algebraic integers, all of whose conjugates lie in a certain compact subset of C, and of classical results on approximation of holomorphic functions by polynomials with integral coefficients. We will try and introduce the relevant geometry of numbers in infinite rank as a means of studying the cohomology of coherents sheaves on these objects. This is joint work with Jean-Benoît Bost.