We will discuss a new approach to prove irrationality of certain periods, including the value at 2 of the Dirichlet L-function associated to the primitive quadratic character with conductor -3. Our method uses rational approximations from the literature and we develop a new framework to make use of these approximations. The key ingredient is an arithmetic holonomy theorem built upon earlier...
I will describe results old and new regarding finiteness and infiniteness of integral points on arithmetic schemes, subject to analytic conditions. I will explain how the language of A-schemes and the use of theta-invariants, as developed in joint work with Bost, provides a geometric framework for this kind of questions, and how it relates to notions of affineness in Arakelov geometry.
