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Quelques questions d'approximation faible pour les tores algébriques

J-L. Colliot-Thélène·V. Suresh
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Abstract

Let K be a global field, T a K-torus and S a finite set of places of K. Let K_v be the completion at a place v. Denote by T(O_v) the maximal compact subgroup of the group T(K_v) of K_v-points of T. We show that the diagonal map from T(K) to the product for v in S of the T(K_v)/T(O_v) need not be onto. As a corollary, for suitable v, the group T(O_v) does not cover all R-equivalence classes in T(K_v). While studying the height zeta functions of an algebraic torus over a function field in one variable over a finite field, D. Bourqui showed that Peyre's constant from the number field case should be multiplied by an integer. The same type of torus as constructed for the above problem enables us to show that this integer need not always be 1. ----- Soient K un corps global, T un K-tore, S un ensemble fini de places de K. On note K_v le complété de K en une place v. Soit T(K), resp. T(K_v), le groupe des points K-rationnels, resp. K_v-rationnels, de T. Notons T(O_v) le sous-groupe compact maximal de T(K_v). Nous montrons que pour T et S convenables l'application diagonale de T(K) vers le produit pour v dans S des T(K_v)/T(O_v) n'est pas surjective. Cela implique que pour v convenable le groupe T(O_v) ne couvre pas forcément toutes les classes de R-équivalence de T(K_v). Lorsque K est un corps de fonctions d'une variable sur un corps fini, en utilisant le même type de tore, nous montrons que le facteur supplémentaire par lequel D. Bourqui doit, lors son étude de la fonction zêta des hauteurs des variétés toriques, multiplier la constante de Peyre, n'est pas toujours égal à 1.

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J-L. Colliot-Thélène, V. Suresh. 2006-06-05. Quelques questions d'approximation faible pour les tores algébriques. https://arxiv.org/abs/math/0606110

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