Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T14:54:50.654Z Has data issue: false hasContentIssue false
  • English
  • Français

CM RELATIONS IN FIBERED POWERS OF ELLIPTIC FAMILIES

Part of: Number theory: Connections with logic Arithmetic algebraic geometry

Published online by Cambridge University Press:  02 August 2017

Fabrizio Barroero Open the ORCID record for Fabrizio Barroero [Opens in a new window]
Fabrizio Barroero*
Affiliation:
Department of Mathematics and Computer Science, University of Basel, Spiegelgasse 1, 4051 Basel, Switzerland (fbarroero@gmail.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

Let $E_{\unicode[STIX]{x1D706}}$ be the Legendre family of elliptic curves. Given $n$ points $P_{1},\ldots ,P_{n}\in E_{\unicode[STIX]{x1D706}}(\overline{\mathbb{Q}(\unicode[STIX]{x1D706})})$, linearly independent over $\mathbb{Z}$, we prove that there are at most finitely many complex numbers $\unicode[STIX]{x1D706}_{0}$ such that $E_{\unicode[STIX]{x1D706}_{0}}$ has complex multiplication and $P_{1}(\unicode[STIX]{x1D706}_{0}),\ldots ,P_{n}(\unicode[STIX]{x1D706}_{0})$ are linearly dependent over End$(E_{\unicode[STIX]{x1D706}_{0}})$. This implies a positive answer to a question of Bertrand and, combined with a previous work in collaboration with Capuano, proves the Zilber–Pink conjecture for a curve in a fibered power of an elliptic scheme when everything is defined over $\overline{\mathbb{Q}}$.

Keywords

unlikely intersectionsZilber–Pink conjectureselliptic curvescomplex multiplication

MSC classification

Primary: 11G05: Elliptic curves over global fields 11G15: Complex multiplication and moduli of abelian varieties
Secondary: 11G50: Heights 11U09: Model theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 18 , Issue 5 , September 2019 , pp. 941 - 956
Copyright
© Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

F. B. was supported by the EPSRC grant EP/N007956/1’ and the SNF grant 165525.

References

André, Y., Shimura varieties, subvarieties and CM points, Six Lectures at the Franco-Taiwan Arithmetic Festival (2001), http://math.cts.nthu.edu.tw/Mathematics/lecnotes/andre2001all.ps.Google Scholar
Barroero, F. and Capuano, L., Linear relations in families of powers of elliptic curves, Algebra Number Theory 10(1) (2016), 195–214.Google Scholar
Bertrand, D., Extensions de D-modules et groupes de Galois différentiels, in p-Adic Analysis (Trento, 1989), Lecture Notes in Mathematics, Volume 1454, pp. 125–141 (Springer, Berlin, 1990).Google Scholar
Bertrand, D., Unlikely intersections in Poincaré biextensions over elliptic schemes, Notre Dame J. Form. Log. 54(3–4) (2013), 365–375.Google Scholar
Bilu, Y., Masser, D. and Zannier, U., An effective ‘theorem of André’ for CM-points on a plane curve, Math. Proc. Cambridge Philos. Soc. 154(1) (2013), 145–152.Google Scholar
Bombieri, E. and Gubler, W., Heights in Diophantine Geometry, New Mathematical Monographs, Volume 4, (Cambridge University Press, Cambridge, 2006).Google Scholar
Breuer, F., Heights of CM points on complex affine curves, Ramanujan J. 5(3) (2001), 311–317.Google Scholar
Colmez, P., Sur la hauteur de Faltings des variétés abéliennes à multiplication complexe, Compos. Math. 111(3) (1998), 359–368.Google Scholar
David, S., Minorations de formes linéaires de logarithmes elliptiques, Mém. Soc. Math. Fr. (N.S.) 62 (1995), iv+143.Google Scholar
David, S., Points de petite hauteur sur les courbes elliptiques, J. Number Theory 64(1) (1997), 104–129.Google Scholar
van den Dries, L., Tame Topology and O-minimal Structures, London Mathematical Society Lecture Note Series, Volume 248 (Cambridge University Press, Cambridge, 1998).Google Scholar
van den Dries, L. and Miller, C., On the real exponential field with restricted analytic functions, Israel J. Math. 85(1–3) (1994), 19–56.Google Scholar
van den Dries, L. and Miller, C., Geometric categories and o-minimal structures, Duke Math. J. 84(2) (1996), 497–540.Google Scholar
Ford, L., Automorphic Functions, second edition (American Mathematical Society, Providence, RI, 1951).Google Scholar
Galateau, A., Une minoration du minimum essentiel sur les variétés abéliennes, Comment. Math. Helv. 85(4) (2010), 775–812.Google Scholar
Habegger, P., Special points on fibered powers of elliptic surfaces, J. Reine Angew. Math. 685 (2013), 143–179.Google Scholar
Habegger, P. and Pila, J., O-minimality and certain atypical intersections, Ann. Sci. Éc. Norm. Supér. (4) 49(4) (2016), 813–858.Google Scholar
Hartshorne, R., Algebraic Geometry, Graduate Texts in Mathematics, Volume 52, (Springer, New York–Heidelberg, 1977).Google Scholar
Lang, S., Elliptic Functions (Addison-Wesley Publishing Co., Inc., Reading, MA–London–Amsterdam, 1973). With an appendix by J. Tate.Google Scholar
Masser, D., Linear relations on algebraic groups, in New Advances in Transcendence Theory (Durham, 1986), pp. 248–262 (Cambridge University Press, Cambridge, 1988).Google Scholar
Masser, D., Counting points of small height on elliptic curves, Bull. Soc. Math. France 117(2) (1989), 247–265.Google Scholar
Masser, D. and Zannier, U., Torsion anomalous points and families of elliptic curves, C. R. Math. Acad. Sci. Paris 346(9–10) (2008), 491–494.Google Scholar
Masser, D. and Zannier, U., Torsion anomalous points and families of elliptic curves, Amer. J. Math. 132(6) (2010), 1677–1691.Google Scholar
Masser, D. and Zannier, U., Torsion points on families of squares of elliptic curves, Math. Ann. 352(2) (2012), 453–484.Google Scholar
Peterzil, Y. and Starchenko, S., Uniform definability of the Weierstrass ℘ functions and generalized tori of dimension one, Selecta Math. (N.S.) 10(4) (2004), 525–550.Google Scholar
Pila, J., Rational points of definable sets and results of André–Oort–Manin–Mumford type, Int. Math. Res. Not. IMRN 13 (2009), 2476–2507.Google Scholar
Pila, J. and Zannier, U., Rational points in periodic analytic sets and the Manin–Mumford conjecture, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19(2) (2008), 149–162.Google Scholar
Pink, R., A common generalization of the conjectures of André–Oort, Manin–Mumford and Mordell–Lang. Manuscript dated 17th April, 2005.Google Scholar
Silverman, J. H., Heights and the specialization map for families of abelian varieties, J. Reine Angew. Math. 342 (1983), 197–211.Google Scholar
Silverman, J. H., The Arithmetic of Elliptic Curves, second edition, Graduate Texts in Mathematics, Volume 106, (Springer, Dordrecht, 2009).Google Scholar
Viada, E., The intersection of a curve with a union of translated codimension-two subgroups in a power of an elliptic curve, Algebra Number Theory 2(3) (2008), 249–298.Google Scholar
Zannier, U., Some Problems of Unlikely Intersections in Arithmetic and Geometry, Annals of Mathematics Studies, Volume 181, (Princeton University Press, 2012). With appendixes by David Masser.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