The hyperring of adèle classes

Abstract

We show that the theory of hyperrings, due to M. Krasner, supplies a perfect framework to understand the algebraic structure of the adele class space of a global field. After promoting F1 to a hyperfield K, we prove that a hyperring of the form R/G (where R is a ring and G is a subgroup of its multiplicative group) is a hyperring extension of K if and only if G union 0 is a subfield of R. This result applies to the adele class space which thus inherits the structure of a hyperring extension H of K. We begin to investigate the content of an algebraic geometry over K. The category of commutative hyperring extensions of K is inclusive of: commutative algebras over fields with semi-linear homomorphisms, abelian groups with injective homomorphisms and a rather exotic land comprising homogeneous non-Desarguesian planes. Finally, we show that for a global field of positive characteristic, the groupoid of the prime elements of the hyperring H is canonically and equivariantly isomorphic to the groupoid of the loops of the maximal abelian cover of the curve associated to the global field.