The hyperring of adele classes.

I present here some recent results (obtained in collaboration with C. Consani [3], [4], [5], [6]) about the "characteristic 1" limit case. The main goal is to prove that the adele class space of a global field, which, up to now, has only been considered as a non-commutative space, has in fact a natural algebraic structure. We will also see that the construction of the Witt ring in characteristic p > 1 has a characteristic 1 analogue and that the deformation of the additive structure implies, in a crucial manner, the entropy function.