L’hyperanneau des classes d’adèles
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.

@article{JTNB_2011__23_1_71_0,
     author = {Alain Connes},
     title = {L{\textquoteright}hyperanneau des classes d{\textquoteright}ad\`eles},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {71--93},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {23},
     number = {1},
     year = {2011},
     doi = {10.5802/jtnb.751},
     zbl = {1277.11091},
     mrnumber = {2780620},
     language = {fr},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.751/}
}
Alain Connes. L’hyperanneau des classes d’adèles. Journal de Théorie des Nombres de Bordeaux, Tome 23 (2011) no. 1, pp. 71-93. doi : 10.5802/jtnb.751. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.751/

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