Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-13T00:44:31.872Z Has data issue: false hasContentIssue false
  • English
  • Français

SPECIAL POINT PROBLEMS WITH ELLIPTIC MODULAR SURFACES

Part of: Arithmetic algebraic geometry Model theory

Published online by Cambridge University Press:  06 November 2013

Jonathan Pila
Jonathan Pila*
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG,U.K.
*
[email protected]
Get access

Abstract

We prove a “special point” result for products of elliptic modular surfaces, elliptic curves, multiplicative groups and complex lines, and deduce a result about vanishing linear combinations of singular moduli and roots of unity.

MSC classification

Secondary: 03C64: Model theory of ordered structures; o-minimality 11G18: Arithmetic aspects of modular and Shimura varieties
Type
Research Article
Information
Mathematika , Volume 60 , Issue 1 , January 2014 , pp. 1 - 31
Copyright
Copyright © University College London 2013 

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.)

References

André, Y., G-Functions and Geometry (Aspects of Mathematics E13), Vieweg (Braunschweig, 1989).CrossRefGoogle Scholar
André, Y., Finititude des couples d’invariants modulaire singuliers sur une courbe algébrique plane non modulaire. J. Reine Angew. Math. 505 (1998), 203–208.Google Scholar
André, Y., Shimura varieties, subvarieties, and CM points. Six lectures at the Franco–Taiwan arithmetic festival (August–September 2001).Google Scholar
Bertrand, D., Unlikely intersections in Poincaré biextensions over elliptic schemes. Notre Dame J. Form. Log. 54 (2013), 365–375.CrossRefGoogle Scholar
Bertrand, D., with an appendix by B. Edixhoven, Special points and Poincaré bi-extensions. Preprint, 2011, arXiv:1104.5178 [math.NT].Google Scholar
Bertrand, D., Masser, D., Pillay, A. and Zannier, U., Relative Manin–Mumford for semi-abelian surfaces. Preprint, 2011, arXiv:1307.1008 [math.NT].Google Scholar
Bilu, Y., Masser, D. and Zannier, U., An effective “Theorem of André” for CM points on plane curves. Math. Proc. Cambridge Philos. Soc. 154 (2013), 145–152.CrossRefGoogle Scholar
Bombieri, E., Masser, D. and Zannier, U., Anomalous subvarieties—structure theorems and applications. Int. Math. Res. Not. 2007 (2007), doi:10.1093/imrn/rnm057.CrossRefGoogle Scholar
David, S., Minorations de formes linéaires de logarithms elliptiques. Mém. Soc. Math. Fr. (N.S.) 62 (1995), 1–143.Google Scholar
Daw, C. and Yafaev, A., An unconditional proof of the André–Oort conjecture for Hilbert modular surfaces. Manuscripta Math. 135 (2011), 263–271.CrossRefGoogle Scholar
van den Dries, L., A generalization of the Tarski–Seidenberg theorem and some non-definability results. Bull. Amer. Math. Soc. 15 (1986), 189–193.CrossRefGoogle Scholar
van den Dries, L., Tame Topology and o-minimal Structures (Lecture Note Series 248), Cambridge University Press (Cambridge, 1998).CrossRefGoogle Scholar
van den Dries, L. and Günaydin, A., The fields of real and complex numbers with a small multiplicative group. Proc. Lond. Math. Soc. (3) 93 (2006), 43–81.Google Scholar
van den Dries, L. and Günaydin, A., Mann pairs. Trans. Amer. Math. Soc. 362 (2010), 2393–2414.Google Scholar
van den Dries, L. and Miller, C., On the real exponential field with restricted analytic functions. Israel J. Math. 85 (1994), 19–56.Google Scholar
Dvornicich, R. and Zannier, U., On sums of roots of unity. Monatsh. Math. 129 (2000), 97–108.CrossRefGoogle Scholar
Evertse, J.-H., The number of solutions of linear equations in roots of unity. Acta Arith. 89 (1999), 45–51.CrossRefGoogle Scholar
Gabrielov, A., Projections of semi-analytic sets. Funct. Anal. Appl. 2 (1968), 282–291.Google Scholar
Habegger, P., Special points on fibred powers of elliptic surfaces. J. Reine Angew. Math.(March 2012), doi:10.1515/crelle-2012-0007.CrossRefGoogle Scholar
Habegger, P. and Pila, J., Some unlikely intersections beyond André–Oort. Compositio Math. 148 (2012), 1–27.Google Scholar
Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers, 5th edn, OUP (Oxford, 1979).Google Scholar
Hindry, M., Autour d’une conjecture de Serge Lang. Invent. Math. 94 (1988), 575–603.CrossRefGoogle Scholar
Klingler, B. and Yafaev, A., The André–Oort conjecture. Ann. of Math. (2) (to appear).Google Scholar
Kühne, L., An effective result of André–Oort type. Ann. of Math. (2) 176 (2012), 651–671.Google Scholar
Laurent, M., Équations diophantiennes exponentielles. Invent. Math. 78 (1984), 299–327.CrossRefGoogle Scholar
Mann, H. B., On linear relations between roots of unity. Mathematika 12 (1965), 107–117.Google Scholar
Masser, D., Small values of the quadratic part of the Néron–Tate height on an abelian variety. Compositio Math. 53 (1984), 153–170.Google Scholar
Masser, D. and Wüstholz, G., Isogeny estimates for abelian varieties, and finiteness results. Ann. of Math. (2) 137 (1993), 459–472.Google Scholar
Oort, F., Canonical lifts and dense sets of CM points. In Arithmetic Geometry, Cortona, 1994 (Symp. Math. XXXVII) (ed. Catanese, F.),Cambridge University Press (Cambridge, 1997), 228–234.Google Scholar
Pellarin, F., Sur une majoration explicite pour un degré d’isogénie liant deux courbes elliptiques. Acta Arith. 100 (2001), 203–243.Google Scholar
Petersson, H., Über die Entwicklungskoeffizienten der automorphen Formen. Acta Math. 58 (1932), 169–215.Google Scholar
Peterzil, Y. and Starchenko, S., Uniform definability of the Weierstrass $\wp $ functions and generalized tori of dimension one. Selecta Math. (N.S.) 10 (2004), 525–550.Google Scholar
Peterzil, Y. and Starchenko, S., Tame complex analysis and o-minimality. Proceedings of the ICM, Hyderabad, 2010.Google Scholar
Pila, J., Modular Ax–Lindemann–Weierstrass with derivatives. Notre Dame J. Form. Log. 54 (2013), 553–565.CrossRefGoogle Scholar
Pila, J., Rational points of definable sets and results of André–Oort–Manin–Mumford type. Int. Math. Res. Not. 2009 (13) (2009), 2476–2507, doi:10.1093/imrn/rnp022.Google Scholar
Pila, J., O-minimality and the André–Oort conjecture for ${ \mathbb{C} }^{n} $. Ann. of Math. (2) 173 (2011), 1779–1840.CrossRefGoogle Scholar
Pila, J. and Tsimerman, J., Ax–Lindemann for ${ \mathcal{A} }_{g} $. Preprint, 2012, arXiv:1206.2663.Google Scholar
Pila, J. and Tsimerman, J., The André–Oort conjecture for the moduli space of abelian surfaces. Compositio Math. 149 (2013), 204–216.CrossRefGoogle Scholar
Pila, J. and Wilkie, A. J., The rational points of a definable set. Duke Math. J. 133 (2006), 591–616.CrossRefGoogle Scholar
Pila, J. and Zannier, U., Rational points in periodic analytic sets and the Manin–Mumford conjecture. Rend. Lincei Mat. Appl. 19 (2008), 149–162.Google Scholar
Pink, R., A combination of the conjectures of Mordell–Lang and André–Oort. In Geometric Methods in Algebra and Number Theory (Progress in Mathematics 253), (eds Bogomolov, F. and Tschinkel, Y.),Birkhäuser (Boston, MA, 2005), 251–282.Google Scholar
Pink, R., A common generalization of the conjectures of André–Oort, Manin–Mumford, and Mordell–Lang, manuscript dated 17 April 2005 available from the author’s webpage.Google Scholar
Raynaud, M., Courbes sur une variété abélienne et points de torsion. Invent. Math. 71 (1) (1983), 207–233.Google Scholar
Raynaud, M., Sous-variétés d’une variété abélienne et points de torsion. In Arithmetic and Geometry, Vol. I (Progress in Mathematics 35), Birkhäuser (Boston, MA, 1983), 327–352.CrossRefGoogle Scholar
Scanlon, T., Automatic uniformity. Int. Math. Res. Not 2004 (62) (2004), 3317–3326, doi:10.1155/S1073792804140816.CrossRefGoogle Scholar
Schinzel, A., Reducibility of lacunary polynomials VIII. Acta Arith. 50 (1988), 91–106.CrossRefGoogle Scholar
Schlickewei, H., Equations in roots of unity. Acta Arith. 76 (1996), 99–108.Google Scholar
Shioda, T., On elliptic modular surfaces. J. Math. Soc. Japan 24 (1972), 20–59.CrossRefGoogle Scholar
Silverberg, A., Torsion points on abelian varieties of CM-type. Compositio Math. 68 (1988), 241–249.Google Scholar
Top, J. and Yui, N., Explicit equations of some elliptic modular surfaces. Rocky Mountain J. Math. 37 (2007), 663–687.Google Scholar
Ullmo, E., Quelques applications du théorème de Ax–Lindemann hyperbolique. Compositio Math.(to appear).Google Scholar
Ullmo, E. and Yafaev, A., Galois orbits and equidistribution of special subvarieties: towards the André–Oort conjecture. Ann. of Math. (2) (to appear).Google Scholar
Ullmo, E. and Yafaev, A., Hyperbolic Ax–Lindemann theorem in the cocompact case. Duke Math. J. (to appear).Google Scholar
Wilkie, A. J., Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Amer. Math. Soc. 9 (1996), 1051–1094.Google Scholar
Zannier, U., On the linear independence of roots of unity over finite extensions of $ \mathbb{Q} $. Acta Arith. 52 (1989), 171–182.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 appendices by D. Masser.Google Scholar
Zilber, B., Exponential sums equations and the Schanuel conjecture. J. Lond. Math. Soc. (2) 65 (2002), 27–44.CrossRefGoogle Scholar