The hyperring of adèle classes

Abstract

Text We show that the theory of hyperrings, due to M. Krasner, supplies a perfect framework to understand the algebraic structure of the adèle class space H K = A K / K × of a global field K . After promoting F 1 to a hyperfield K, we prove that a hyperring of the form R / G (where R is a ring and G ⊂ R × is a subgroup of its multiplicative group) is a hyperring extension of K if and only if G ∪ { 0 } is a subfield of R. This result applies to the adèle class space which thus inherits the structure of a hyperring extension H K 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 K of positive characteristic, the groupoid of the prime elements of the hyperring H K is canonically and equivariantly isomorphic to the groupoid of the loops of the maximal abelian cover of the curve associated to the global field K . Video For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=3LSKD_PfJyc.