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Sur la conjecture de Manin pour certaines surfaces de Châtelet

Part of: Diophantine equations

Published online by Cambridge University Press:  08 March 2013

Régis de la Bretèche  and
Gérald Tenenbaum
Régis de la Bretèche
Affiliation:
Institut de Mathématiques de Jussieu, UMR 7586, Université Paris Diderot-Paris 7, UPR de Mathématiques, case 7012, Bâtiment Chevaleret, 75205 Paris Cedex 13, France ([email protected])
Gérald Tenenbaum
Affiliation:
Institut Élie Cartan, Université de Lorraine, BP 70239, 54506 Vandœuvre-lès-Nancy Cedex, France ([email protected])
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Résumé

Nous démontrons, sous la forme forte conjecturée par Peyre, la conjecture de Manin pour les surfaces de Châtelet dont les équations sont du type ${y}^{2} + {z}^{2} = P(x, 1)$, où $P$ est une forme binaire quartique à coefficients entiers irréductible sur $ \mathbb{Q} [i] $ ou produit de deux formes quadratiques à coefficients entiers irréductibles sur $ \mathbb{Q} [i] $. De plus, nous fournissons une estimation explicite du terme d’erreur de la formule asymptotique sous-jacente. Cela finalise essentiellement la validation de la conjecture de Manin pour l’ensemble des surfaces de Châtelet. La preuve s’appuie sur deux méthodes nouvelles, concernant, du part, les estimations en moyenne d’oscillations locales de caractères sur les diviseurs, et, d’autre part, les majorations de certaines fonctions arithmétiques de formes binaires.

Abstract

We prove Manin’s conjecture, in the strong form conjectured by Peyre, for Châtelet surfaces associated to surfaces of the type ${y}^{2} + {z}^{2} = P(x, 1)$, where $P$ is a binary quartic form with integer coefficients that is either irreducible over $ \mathbb{Q} [i] $ or the product of two quadratic forms with integer coefficients and irreducible over $ \mathbb{Q} [i] $. Moreover, we provide an explicit upper bound for the remainder term in the relevant asymptotic formula. This essentially settles Manin’s conjecture for all Châtelet surfaces. The proof rests on two new tools, namely upper bounds for mean values of local oscillations of characters on divisors and sharp upper estimates for mean values of arithmetic functions of binary forms.

Keywords

Châtelet surfacesManin’s conjecturePeyre’s constantdel Pezzo surfaces of degree 4count of lattice points over algebraic varietiesnumber of representations as a sum of two squaresmethod of descent

MSC classification

Secondary: 11D45: Counting solutions of Diophantine equations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 12 , Issue 4 , October 2013 , pp. 759 - 819
Copyright
©Cambridge University Press 2013 

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