It is well-known from Weil that the isomorphism classes of rank n vector bundles on an algebraic curve can be written as the set of certain double cosets of GL(n,A), where A is the adeles of the curve. I will introduce such presentation in the realm of algebraic geometry and discuss two of its applications: the first is the Tamagawa number formula (proved by Gaitsgory-Lurie), and the second is the Verlinde formula in positive characteristic