Skip to main content Accessibility help
×
Home
Hostname: page-component-77ffc5d9c7-5j57r Total loading time: 0.258 Render date: 2021-04-23T03:14:18.814Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Sur la conjecture de Manin pour certaines surfaces de Châtelet

Published online by Cambridge University Press:  08 March 2013

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 (breteche@math.jussieu.fr)
Gérald Tenenbaum
Affiliation:
Institut Élie Cartan, Université de Lorraine, BP 70239, 54506 Vandœuvre-lès-Nancy Cedex, France (gerald.tenenbaum@univ-lorraine.fr)

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.

Type
Research Article
Copyright
©Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below.

References

de la Bretèche, R. et Browning, T. D., Sums of arithmetic functions over values of binary forms, Acta Arith. 125 (3) (2006), 291304.CrossRefGoogle Scholar
de la Bretèche, R. et Browning, T. D., Binary linear forms as sums of two squares, Compositio Math. 144 (6) (2008), 13751402.CrossRefGoogle Scholar
de la Bretèche, R. et Browning, T. D., Le problème des diviseurs pour des formes binaires de degré 4, J. Reine Angew. Math. 646 (2010), 144.CrossRefGoogle Scholar
de la Bretèche, R. et Browning, T. D., Binary forms of two squares and Châtelet surfaces, Israel J. Math. 191 (2012), 9731012.CrossRefGoogle Scholar
de la Bretèche, R., Browning, T. D. et Peyre, E., On Manin’s conjecture for a family of Châtelet surfaces, Ann. of Math. 175 (2012), 147.CrossRefGoogle Scholar
de la Bretèche, R. et Tenenbaum, G., Oscillations localisées sur les diviseurs, J. Lond. Math. Soc. 85 (3) (2012), 669693.CrossRefGoogle Scholar
de la Bretèche, R. et Tenenbaum, G., Moyennes de fonctions arithmétiques de formes binaires, Mathematika 58 (2012), 290304.CrossRefGoogle Scholar
Browning, T. D., Quantitative arithmetic of projective varieties, Prog. Math., Volume 277 (Birkhäuser, 2009).CrossRefGoogle Scholar
Browning, T. D., Linear growth for Châtelet surfaces, Math. Ann. 346 (2010), 4150.CrossRefGoogle Scholar
Colliot-Thélène, J.-L., Coray, D. et Sansuc, J.-J., Descente et principe de Hasse pour certaines variétés rationnelles, J. Reine Angew. Math. 320 (1980), 150191.Google Scholar
Colliot-Thélène, J.-L. et Sansuc, J.-J., La descente sur les variétés rationnelles, Journées de géométrie algébrique d’Angers (1979) (ed. Beauville, A.), pp. 223237 (Sijthoff & Noordhoff, Alphen aan den Rijn, 1980).Google Scholar
Colliot-Thélène, J.-L., Sansuc, J.-J. et Swinnerton-Dyer, P., Intersections of two quadrics and Châtelet surfaces. I, J. Reine Angew. Math. 373 (1987), 37107.Google Scholar
Colliot-Thélène, J.-L., Sansuc, J.-J. et Swinnerton-Dyer, P., Intersections of two quadrics and Châtelet surfaces. II, J. Reine Angew. Math. 374 (1987), 72168.Google Scholar
Cook, R. J., Simultaneous quadratic equations, J. Lond. Math. Soc. 4 (2) (1971), 319326.CrossRefGoogle Scholar
Daniel, S., On the divisor-sum problem for binary forms, J. Reine Angew. Math. 507 (1999), 107129.CrossRefGoogle Scholar
Derenthal, U., On a constant arising in Manin’s conjecture for del Pezzo surfaces, Math. Res. Lett. 14 (3) (2007), 481489.CrossRefGoogle Scholar
Hall, R. R. et Tenenbaum, G., Divisors, Cambridge tracts in mathematics, Volume 90, Cambridge University Press (1988, paperback ed. 2008).Google Scholar
Heath-Brown, D. R., Diophantine approximation with square-free numbers, Math. Z. 187 (1984), 335344.CrossRefGoogle Scholar
Heath-Brown, D. R., Linear relations amongst sums of two squares, in Number theory and algebraic geometry, London Math. Soc. Lecture Note Ser., Volume 303, pp. 133176 (CUP, 2003).Google Scholar
Heilbronn, H., Zeta-functions and $L$-functions, in Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pp. 204230 (Thompson, Washington, D.C, 1967).Google Scholar
Henriot, K., Nair–Tenenbaum bounds uniform with respect to the discriminant, Math. Proc. Cambridge Philos. Soc. 152 (2012), 405424.CrossRefGoogle Scholar
Hooley, C., A new technique and its applications to the theory of numbers, Proc. Lond. Math. Soc. 38 (3) (1979), 115151.CrossRefGoogle Scholar
Iskovskih, V. A., A counterexample to the Hasse principle for systems of two quadratic forms in five variables, Mat. Z. 10 (1971), 253257, English transl. in Math. Notes 10 (1971), 575–577.Google Scholar
Maier, H. et Tenenbaum, G., On the normal concentration of divisors, 2, Math. Proc. Cambridge Philos. Soc. 147 (3) (2009), 593614.CrossRefGoogle Scholar
Marasingha, G., On the representation of almost primes by pairs of quadratic forms, Acta Arith. 124 (4) (2006), 327355.CrossRefGoogle Scholar
Nagell, T., Introduction to number theory, 2nd edn (Chelsea, 1964).Google Scholar
Peyre, E., Hauteurs et mesures de Tamagawa sur les variétés de Fano, Duke Math. J. 79 (1995), 101218.CrossRefGoogle Scholar
Peyre, E., Points de hauteur bornée, topologie adélique et mesures de Tamagawa, J. Théor. Nombres Bordeaux 15 (1) (2003), 319349.CrossRefGoogle Scholar
Skorobogatov, A., Torsors and rational points, Cambridge Tracts in Mathematics, Volume 144 (Cambridge University Press, Cambridge, 2001), viii+187 pp.CrossRefGoogle Scholar
Stewart, C. L., On the number of solutions of polynomial congruences and Thue equations, J. Amer. Math. Soc. 4 (1991), 793835.CrossRefGoogle Scholar
Tenenbaum, G., Sur une question d’Erdős et Schinzel, II, Invent. Math. 99 (1990), 215224.CrossRefGoogle Scholar
Tenenbaum, G., Introduction à la théorie analytique et probabiliste des nombres, troisième édition, Belin (coll. Échelles, 2008), 592 pp.Google Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 50 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 23rd April 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Sur la conjecture de Manin pour certaines surfaces de Châtelet
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Sur la conjecture de Manin pour certaines surfaces de Châtelet
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Sur la conjecture de Manin pour certaines surfaces de Châtelet
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *