Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-26T14:30:09.557Z Has data issue: false hasContentIssue false
  • English
  • Français

THE EXPLICIT MORDELL CONJECTURE FOR FAMILIES OF CURVES

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  19 September 2019

SARA CHECCOLI Open the ORCID record for SARA CHECCOLI [Opens in a new window] ,
FRANCESCO VENEZIANO Open the ORCID record for FRANCESCO VENEZIANO [Opens in a new window]  and
EVELINA VIADA
SARA CHECCOLI
Affiliation:
Institut Fourier, 100 rue des Maths, BP74 38402 Saint-Martin-d’Hères Cedex, France; sara.checcoli@univ-grenoble-alpes.fr
FRANCESCO VENEZIANO
Affiliation:
Collegio Puteano, Scuola Normale Superiore, Piazza dei Cavalieri, 3, I-56100 Pisa, Italy; francesco.veneziano@sns.it
EVELINA VIADA
Affiliation:
Mathematisches Institut, Georg-August-Universität, Bunsenstraße 3-5, D-37073, Göttingen, Germany; evelina.viada@math.ethz.ch

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this article we prove the explicit Mordell Conjecture for large families of curves. In addition, we introduce a method, of easy application, to compute all rational points on curves of quite general shape and increasing genus. The method bases on some explicit and sharp estimates for the height of such rational points, and the bounds are small enough to successfully implement a computer search. As an evidence of the simplicity of its application, we present a variety of explicit examples and explain how to produce many others. In the appendix our method is compared in detail to the classical method of Manin–Demjanenko and the analysis of our explicit examples is carried to conclusion.

MSC classification

Primary: 11G50: Heights
Secondary: 14G40: Arithmetic varieties and schemes; Arakelov theory; heights
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e31
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

References

LMFDB - The $L$ -functions and Modular Forms Database, http://www.lmfdb.org/.Google Scholar
Bombieri, E. and Gubler, W., Heights in Diophantine Geometry, New Mathematical Monographs, 4 (Cambridge University Press, Cambridge, 2006).Google Scholar
Bombieri, E., Masser, D. and Zannier, U., ‘Anomalous subvarieties—structure theorems and applications’, Int. Math. Res. Not. IMRN 19 (2007), Art. ID rnm057, 33.Google Scholar
Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system. I. The user language’, J. Symbolic Comput. 24(3–4) (1997), 235–265. See also the Magma home page at http://magma.maths.usyd.edu.au/magma/.Google Scholar
Bost, J.-B., Gillet, H. and Soulé, C., ‘Heights of projective varieties and positive Green forms’, J. Amer. Math. Soc. 7(4) (1994), 903–1027.Google Scholar
Bruin, P., ‘Bornes optimales pour la différence entre la hauteur de Weil et la hauteur de Néron-Tate sur les courbes elliptiques sur ℚ’, Acta Arith. 160(4) (2013), 385–397.Google Scholar
Bruin, N. and Stoll, M., ‘The Mordell–Weil sieve: proving non-existence of rational points on curves’, LMS J. Comput. Math. 13 (2010), 272–306.Google Scholar
Chabauty, C., ‘Sur les points rationnels des courbes algébriques de genre supérieur à l’unité’, C. R. Math. Acad. Sci. Paris 212 (1941), 882–885.Google Scholar
Checcoli, S., Veneziano, F. and Viada, E., ‘On the explicit Torsion Anomalous Conjecture’, Trans. Amer. Math. Soc. 369 (2017), 6465–6491.Google Scholar
Coleman, R. F., ‘Effective Chabauty’, Duke Math. J. 52(3) (1985), 765–770.Google Scholar
Cremona, J. E., Elliptic Curve Data, https://johncremona.github.io/ecdata/.Google Scholar
Cremona, J. E., Prickett, M. and Siksek, S., ‘Height difference bounds for elliptic curves over number fields’, J. Number Theory 116 (2006), 42–68.Google Scholar
Demjanenko, V. A., ‘Rational points of a class of algebraic curves’, Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966), 1373–1396.Google Scholar
Faltings, G., ‘Endlichkeitssätze für abelsche Varietäten über Zahlkörpern’, Invent. Math. 73(3) (1983), 349–366.Google Scholar
Faltings, G., ‘Diophantine approximation on abelian varieties’, Ann. of Math. (2) 133(3) (1991), 549–576.Google Scholar
Faltings, G., ‘The general case of S. Lang’s conjecture’, inBarsotti Symposium in Algebraic Geometry (Abano Terme, 1991), (eds. Christante, V. and Messing, W.) Perspectives in Math. 15 (Academic Press, San Diego, CA, 1994), 175–182.Google Scholar
Flynn, E. V., ‘A flexible method for applying Chabauty’s theorem’, Compos. Math. 105(1) (1997), 79–94.Google Scholar
Fulton, W., Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 2 (Springer, Berlin, 1984).Google Scholar
Girard, M. and Kulesz, L., ‘Computation of sets of rational points of genus-3 curves via the Dem’janenko-Manin method’, LMS J. Comput. Math. 8 (2005), 267–300.Google Scholar
Habegger, P., ‘Intersecting subvarieties of 𝔾m n with algebraic subgroups’, Math. Ann. 342(2) (2008), 449–466.Google Scholar
Hindry, M., ‘Autour d’une conjecture de Serge Lang’, Invent. Math. 94(3) (1988), 575–603.Google Scholar
Hindry, M. and Silverman, J. H., Diophantine Geometry, Graduate Texts in Mathematics, 201 (Springer, New York, 2000), An introduction.Google Scholar
Kappe, L.-C. and Warren, B., ‘An elementary test for the Galois group of a quartic polynomial’, Amer. Math. Monthly 96(2) (1989), 133–137.Google Scholar
Kulesz, L., ‘Application de la méthode de Dem’janenko-Manin à certaines familles de courbes de genre 2 et 3’, J. Number Theory 76(1) (1999), 130–146.Google Scholar
Kulesz, L., Matera, G. and Schost, E., ‘Uniform bounds on the number of rational points of a family of curves of genus 2’, J. Number Theory 108(2) (2004), 241–267.Google Scholar
Manin, Ju. I., ‘The p-torsion of elliptic curves is uniformly bounded’, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 459–465.Google Scholar
Masser, D. W. and Wüstholz, G., ‘Estimating isogenies on elliptic curves’, Invent. Math. 100(1) (1990), 1–24.Google Scholar
Mazur, B., ‘Modular curves and the Eisenstein ideal’, Publ. Math. Inst. Hautes Études Sci. 47 (1978), 33–186. 1977.Google Scholar
McCallum, W. and Poonen, B., ‘The Method of Chabauty and Coleman’, Explicit methods in number theory; rational points and diophantine equations, Panoramas et Synthèses 36 (Société Math. de France, 2012), 99–117.Google Scholar
Mordell, L. J., ‘On the rational solutions of the indeterminate equation of the third and fourth degrees’, Math. Proc. Cambridge Philos. Soc. 21 (1922), 179–192.Google Scholar
Parent, P., ‘Bornes effectives pour la torsion des courbes elliptiques sur les corps de nombres’, J. Reine Angew. Math. 506 (1999), 85–116.Google Scholar
Philippon, P., ‘Sur des hauteurs alternatives. I’, Math. Ann. 289(2) (1991), 255–283.Google Scholar
Philippon, P., ‘Sur des hauteurs alternatives. III.’, J. Math. Pures Appl. (9) 74(4) (1995), 345–365.Google Scholar
Philippon, P., ‘Sur une question d’orthogonalité dans les puissances de courbes elliptiques’. Preprint, 2012, arXiv:hal–00801376.Google Scholar
Rémond, G., ‘Décompte dans une conjecture de Lang’, Invent. Math. 142(3) (2000), 513–545.Google Scholar
Serre, J.-P., Lectures on the Mordell–Weil Theorem, Aspects of Mathematics, E15 (Friedr. Vieweg & Sohn, Braunschweig, 1989), Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt.Google Scholar
Siksek, S., ‘Explicit Chabauty over number fields’, Algebra Number Theory 7(4) (2013), 765–793.Google Scholar
Silverman, J. H., The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106 (Springer, New York, 1986).Google Scholar
Silverman, J. H., ‘Rational points on certain families of curves of genus at least 2’, Proc. Lond. Math. Soc. (3) 55(3) (1987), 465–481.Google Scholar
Silverman, J. H., ‘The difference between the Weil height and the canonical height on elliptic curves’, Math. Comp. 55(192) (1990), 723–743.Google Scholar
Silverman, J. H., ‘Computing rational points on rank 1 elliptic curves via L-series and canonical heights’, Math. Comp. 68(226) (1999), 835–858.Google Scholar
Stoll, M., ‘Rational points on curves’, J. Théor. Nombres Bordeaux 23(1) (2011), 257–277.Google Scholar
The PARI Group. PARI/GP version 2.8.0. 2015. http://pari.math.u-bordeaux.fr/.Google Scholar
Viada, E., ‘An explicit Manin–Dem’janenko theorem in elliptic curves’, Canad. J. Math. 70(5) (2018), 1173–1200.Google Scholar
Viada, E., ‘The intersection of a curve with algebraic subgroups in a product of elliptic curves’, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2(1) (2003), 47–75.Google Scholar
Viada, E., ‘Explicit height bounds and the effective Mordell–Lang conjecture’, Riv. Mat. Uni. Parma (8) 7(1) (2016), 101–131. Proceedings of the ‘Third Italian Number Theory Meeting’ Pisa (Italy), September 21–24, 2015.Google Scholar
Vojta, P., ‘Siegel’s theorem in the compact case’, Ann. of Math. (2) 133(3) (1991), 509–548.Google Scholar
Zannier, U., Some Problems of Unlikely Intersections in Arithmetic and Geometry, Annals of Mathematics Studies, 181 (Princeton University Press, Princeton, NJ, 2012), With appendixes by David Masser.Google Scholar
Zhang, S., ‘Positive line bundles on arithmetic varieties’, J. Amer. Math. Soc. 8(1) (1995), 187–221.Google Scholar
Zimmer, H. G., ‘On the difference of the Weil height and the Néron-Tate height’, Math. Z. 147(1) (1976), 35–51.Google Scholar