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Some unlikely intersections beyond André–Oort

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry Abelian varieties and schemes Model theory

Published online by Cambridge University Press:  30 August 2011

P. Habegger  and
J. Pila
P. Habegger
Affiliation:
Institut fuer Mathematik, Goethe Universitaet Frankfurt, Robert-Mayer-Strasse 6-8, 60325 Frankfurt am Main, Germany (email: habegger@math.uni-frankfurt.ch)
J. Pila
Affiliation:
School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK (email: pila@maths.ox.ac.uk)
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Abstract

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According to the André–Oort conjecture, an algebraic curve in Y (1)n that is not equal to a special subvariety contains only finitely many points which correspond to ann-tuple of elliptic curves with complex multiplication. Pink’s conjecture generalizes the André–Oort conjecture to the extent that if the curve is not contained in a special subvariety of positive codimension, then it is expected to meet the union of all special subvarieties of codimension two in only finitely many points. We prove this for a large class of curves in Y (1)n. When restricting to special subvarieties of codimension two that are not strongly special we obtain finiteness for all curves defined over . Finally, we formulate and prove a variant of the Mordell–Lang conjecture for subvarieties of Y (1)n.

Keywords

André–Oort conjectureZilber–Pink conjectureunlikely intersection

MSC classification

Secondary: 11G18: Arithmetic aspects of modular and Shimura varieties 03C64: Model theory of ordered structures; o-minimality 11G15: Complex multiplication and moduli of abelian varieties 11G50: Heights 14G35: Modular and Shimura varieties 14K22: Complex multiplication
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 1 , January 2012 , pp. 1 - 27
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[And92]André, Y., Mumford–Tate groups of mixed Hodge structures and the theorem of the fixed part, Compositio Math. 82 (1992), 1–24.Google Scholar
[And98]André, Y., Finitude des couples d’invariants modulaires singuliers sur une courbe algébrique plane non modulaire, J. Reine Angew. Math. 505 (1998), 203–208.CrossRefGoogle Scholar
[Ax71]Ax, J., On Schanuel’s conjectures, Ann. of Math. (2) 93 (1971), 252–268.CrossRefGoogle Scholar
[Bak75]Baker, A., Transcendental number theory (Cambridge University Press, Cambridge, 1975).CrossRefGoogle Scholar
[BG06]Bombieri, E. and Gubler, W., Heights in diophantine geometry (Cambridge University Press, Cambridge, 2006).Google Scholar
[BMZ08]Bombieri, E., Masser, D. and Zannier, U., On unlikely intersections of complex varieties with tori, Acta Arith. 133 (2008), 309–323.CrossRefGoogle Scholar
[Bre01]Breuer, F., Heights of CM points on complex affine curves, Ramanujan J. 5 (2001), 311–317.CrossRefGoogle Scholar
[Cas57]Cassels, J. W.  S., An introduction to diophantine approximation (Cambridge University Press, Cambridge, 1957).Google Scholar
[Cox89]Cox, D. A., Primes of the form x 2+ny 2 (John Wiley & Sons, New York, 1989).Google Scholar
[Dav00]Davenport, H., Multiplicative number theory (Springer, Berlin, 2000).Google Scholar
[Dav95]David, S., Minorations de formes linéaires de logarithmes elliptiques, Mém. Soc. Math. Fr. (N.S.) 62 (1995), 1–143.Google Scholar
[Del71]Deligne, P., Théorie de Hodge: II, Publ. Math. Inst. Hautes Études Sci. 40 (1971), 5–57.CrossRefGoogle Scholar
[Dri84]van den Dries, L., Remarks on Tarski’s problem concerning (R,+,⋅,exp ), in Logic colloquium ’82 (Florence, 1982), Studies in Logic and the Foundations of Mathematics, vol. 112 (North-Holland, Amsterdam, 1984), 97–121.Google Scholar
[Dri86]van den Dries, L., A generalization of the Tarski–Seidenberg theorem, and some nondefinability results, Bull. Amer. Math. Soc. (N.S.) 15 (1986), 189–193.CrossRefGoogle Scholar
[Dri98]van den Dries, L., Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248 (Cambridge University Press, Cambridge, 1998).CrossRefGoogle Scholar
[DMM94]van den Dries, L., Macintyre, A. and Marker, D., The elementary theory of restricted analytic fields with exponentiation, Ann. of Math. (2) 140 (1994), 183–205.CrossRefGoogle Scholar
[DM94]van den Dries, L. and Miller, C., On the real exponential field with restricted analytic functions, Israel J. Math. 85 (1994), 19–56.CrossRefGoogle Scholar
[DM96]van den Dries, L. and Miller, C., Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), 497–540.Google Scholar
[Edi98]Edixhoven, B., Special points on the product of two modular curves, Compositio Math. 114 (1998), 315–328.CrossRefGoogle Scholar
[Edi05]Edixhoven, B., Special points on products of modular curves, Duke Math. J. 126 (2005), 325–348.CrossRefGoogle Scholar
[EMOT81]Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., Higher transcendental functions, Vol. I (Robert E. Krieger, Melbourne, FL, 1981).Google Scholar
[Gab68]Gabrièlov, A. M., Projections of semianalytic sets, Funktsional. Anal. i Prilozhen. 2 (1968), 18–30.CrossRefGoogle Scholar
[GS00]Granville, A. and Stark, H. M., ABC implies no ‘Siegel zeros’ for L-functions of characters with negative discriminant, Invent. Math. 139 (2000), 509–523.CrossRefGoogle Scholar
[GR84]Grauert, H. and Remmert, R., Coherent analytic sheaves (Springer, Berlin, 1984).CrossRefGoogle Scholar
[Hab07]Habegger, P., Heights and multiplicative relations on algebraic varieties, PhD thesis, University of Basel (2007).Google Scholar
[Hab10]Habegger, P., Weakly bounded height on modular curves, Acta Math. Vietnam. 35 (2010), 43–69.Google Scholar
[Hus04]Husemöller, D., Elliptic curves (Springer, Berlin, 2004).Google Scholar
[Ima76]Imai, H., On the Hodge groups of some abelian varieties, Kōdai Math. Semin. Rep. 27 (1976), 367–372.Google Scholar
[Kol68]Kolchin, E. R., Algebraic groups and algebraic dependence, Amer. J. Math. 90 (1968), 1151–1164.CrossRefGoogle Scholar
[Lan87]Lang, S., Elliptic functions (Springer, Berlin, 1987).CrossRefGoogle Scholar
[MW90]Masser, D. W. and Wüstholz, G., Estimating isogenies on elliptic curves, Invent. Math. 100 (1990), 1–24.CrossRefGoogle Scholar
[Mau08]Maurin, G., Courbes algébriques et équations multiplicatives, Math. Ann. 341 (2008), 789–824.CrossRefGoogle Scholar
[Moo98]Moonen, B., Linearity properties of Shimura varieties. I, J. Algebraic Geom. 7 (1998), 539–567.Google Scholar
[NT91]Nakkajima, Y. and Taguchi, Y., A generalization of the Chowla–Selberg formula, J. Reine Angew. Math. 419 (1991), 119–124.Google Scholar
[Pel01]Pellarin, F., Sur une majoration explicite pour un degré d’isogénie liant deux courbes elliptiques, Acta Arith. 100 (2001), 203–243.CrossRefGoogle Scholar
[PS04]Peterzil, Y. and Starchenko, S., Uniform definability of the Weierstrass ℘ functions and generalized tori of dimension one, Selecta Math. (N.S.) 10 (2004), 525–550.CrossRefGoogle Scholar
[Pil09a]Pila, J., On the algebraic points of a definable set, Selecta Math. (N.S.) 15 (2009), 151–170.CrossRefGoogle Scholar
[Pil09b]Pila, J., Rational points of definable sets and results of André–Oort–Manin–Mumford type, Int. Math. Res. Not. IMRN (2009), 2476–2507.Google Scholar
[Pil11]Pila, J., O-minimality and the André–Oort conjecture for , Ann. of Math. (2) 173 (2011), 1779–1840.CrossRefGoogle Scholar
[PW06]Pila, J. and Wilkie, A. J., The rational points of a definable set, Duke Math. J. 133 (2006), 591–616.CrossRefGoogle Scholar
[PZ08]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 (2008), 149–162.Google Scholar
[PS86]Pillay, A. and Steinhorn, C., Definable sets in ordered structures. I, Trans. Amer. Math. Soc. 295 (1986), 565–592.CrossRefGoogle Scholar
[Pin05]Pink, R., A common generalization of the conjectures of André–Oort, Manin–Mumford, and Mordell–Lang, Preprint (2005), available at www.math.ethz.ch/∼pink/ftp/ (AOMMML.pdf).Google Scholar
[Ray85]Raynaud, M., Hauteurs et isogénies, in Seminar on arithmetic bundles: the Mordell conjecture (Paris, 1983/84), Astérisque 127 (1985), 199–234.Google Scholar
[Ser72]Serre, J.-P., Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259–331.CrossRefGoogle Scholar
[UY09]Ullmo, E. and Yafaev, A., The André–Oort conjecture for products of modular curves, in Arithmetic geometry, Clay Mathematics Proceedings, vol. 8 (American Mathematical Society, Providence, RI, 2009), 431–439.Google Scholar
[Wil96]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.CrossRefGoogle Scholar
[Zil02]Zilber, B., Exponential sums equations and the Schanuel conjecture, J. Lond. Math. Soc. (2) 65 (2002), 27–44.CrossRefGoogle Scholar