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MOYENNES DE FONCTIONS ARITHMÉTIQUES DE FORMES BINAIRES

Published online by Cambridge University Press:  19 January 2012

Régis de la Bretèche
Affiliation:
Institut de Mathématiques de Jussieu, UMR 7586, Université Paris Diderot-Paris 7, UFR de Mathématiques, case 7012, Bâtiment Chevaleret, 75205 Paris Cedex 13, France (email: breteche@math.jussieu.fr)
Gérald Tenenbaum
Affiliation:
Institut Élie Cartan, Université de Nancy 1, BP 239, 54506 Vandœuvre Cedex, France (email: gerald.tenenbaum@iecn.u-nancy.fr)

Abstract

Extending classical results of Nair and Tenenbaum, we provide general, sharp upper bounds for sums of the type where x,y,u,v have comparable logarithms, F belongs to a class defined by a weak form of sub-multiplicativity, and the Qj are arbitrary binary forms. A specific feature of the results is that the bounds are uniform within the F-class and that, as in a recent version given by Henriot, the dependency with respect to the coefficients of the Qj is made explicit. These estimates play a crucial rôle in the proof, published separately by the authors, of Manin’s conjecture for Châtelet surfaces.

Résumé

Généralisant des résultats classiques de Nair et Tenenbaum, nous fournissons des majorations générales et optimales pour des sommes du type où les paramètres x,y,u,v ont des logarithmes comparables, F décrit une classe de fonctions définie par une condition de sous-multiplicativité faible, et les Qj sont des formes binaires arbitraires. Ces résultats sont caractérisés par leur uniformité dans la classe des fonctions F et, à l’instar d’une version récente donnée par Henriot, une dépendance explicite en fonction des coefficients des Qj. Ces estimations jouent un rôle crucial dans la preuve, publiée séparément par les auteurs, de la conjecture de Manin pour les surfaces de Châtelet.

Type
Research Article
Copyright
Copyright © University College London 2012

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References

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MOYENNES DE FONCTIONS ARITHMÉTIQUES DE FORMES BINAIRES
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