L’hyperanneau des classes d’adèles

Connes, Alain

Journal de théorie des nombres de Bordeaux, Tome 23 (2011) no. 1, pp. 71-93.

J’exposerai ici quelques résultats récents (obtenus en collaboration avec C. Consani [3], [4], [5], [6]) qui portent sur le cas limite de la “caractéristique $1$”. Le but principal est de montrer que l’espace des classes d’adèles d’un corps global, qui jusqu’à présent n’a été considéré que comme un espace (non-commutatif), admet en fait une structure algébrique naturelle. Nous verrons également que la construction de l’anneau de Witt d’un anneau de caractéristique $p>1$ admet un analogue en caractéristique $1$ et que la déformation de la structure additive implique de manière cruciale l’entropie.

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 adèle 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.

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