On the number of solutions of polynomial congruences and Thue equations

International audienceWe 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 Q[i] or the product of two quadratic forms with integer coefficients and irreducible over 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. Pezzo surfaces of degree 4, count of lattice points over algebraic varieties, number of representations as a sum of two squares, method of descent.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 Q[i] ou produit de deux formes quadratiques à coefficients entiers irréductibles sur 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. Mots clefs : Surfaces de Châtelet , conjecture de Manin, constante de Peyre, surfaces de del Pezzo de degré 4, comptage de points entiers sur des variétés algébriques , nombre de représentations comme somme de deux carrés , méthode de descente

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“…Démonstration. L'inégalité T (p ν ) dp ((1−1/d)ν) est dueà Stewart [30], les autres sont etablies dans le livre de Nagell -cf. [ …”

Section: Sur Le Nombre De Solutions Locales D'uneéquation Polynomialeunclassified

Sur la conjecture de Manin pour certaines surfaces de Châtelet

Bretèche¹,

Tenenbaum²

2013

J. Inst. Math. Jussieu

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“…We will follow Stewart's [14] results for Thue inequalities with arbitrary degree and sharpen them for quartic Thue equations.…”

Section: Small Solutionsmentioning

confidence: 92%

“…In this paper we will give upper bounds for the number of integral solutions to (1) with large discriminant and no restriction on J . We will use some ideas of Stewart [14] to prove Theorem 1.1. Let F (x, y) be an irreducible binary form with integral coefficients and degree 4.…”

Section: Introductionmentioning

confidence: 99%

“…We also combine some analytic methods from [14] with some geometric methods from [12] to show that Theorem 1.2. Let F (x, y) be an irreducible binary form with integral coefficients and degree 4 that splits in R. Then the Diophantine equation (1) has at most 37 solutions in integers x and y (with (x, y) and (−x, −y) regarded as the same), provided that the discriminant of F is greater than D 0 , where D 0 is an effectively computable constant.…”

Section: Introductionmentioning

confidence: 99%

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Quartic Thue equations

Akhtari

Okazaki

2010

Journal of Number Theory

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“…(T ) has been called the Thue equation, and the Thue result has been strengthened by a number of authors. For the story of estimating the number N c (F n , h) of all coprime solutions of (T ) we refer the reader to two papers of Stewart [8,9]. (We recall that two integers a, b with ab = 0 are coprime if (a, b) = 1, where the symbol (a, b) denotes GCD of a, b.…”

Section: Introductionmentioning

confidence: 99%