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Applications of the hyperbolic Ax–Schanuel conjecture

Part of: Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry

Published online by Cambridge University Press:  13 August 2018

Christopher Daw  and
Jinbo Ren
Christopher Daw
Affiliation:
Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 217, Reading, Berkshire RG6 6AH, UK email chris.daw@reading.ac.uk
Jinbo Ren
Affiliation:
Institut des Hautes Études Scientifiques, Le Bois-Marie 35, route de Chartres, 91440 Bures-sur-Yvette, France email renjinbo@ihes.fr
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Abstract

In 2014, Pila and Tsimerman gave a proof of the Ax–Schanuel conjecture for the$j$-function and, with Mok, have recently announced a proof of its generalization to any (pure) Shimura variety. We refer to this generalization as the hyperbolic Ax–Schanuel conjecture. In this article, we show that the hyperbolic Ax–Schanuel conjecture can be used to reduce the Zilber–Pink conjecture for Shimura varieties to a problem of point counting. We further show that this point counting problem can be tackled in a number of cases using the Pila–Wilkie counting theorem and several arithmetic conjectures. Our methods are inspired by previous applications of the Pila–Zannier method and, in particular, the recent proof by Habegger and Pila of the Zilber–Pink conjecture for curves in abelian varieties.

Keywords

hyperbolic Ax–Schanuel conjectureZilber–Pink conjectureShimura varieties

MSC classification

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties 14G35: Modular and Shimura varieties
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 9 , September 2018 , pp. 1843 - 1888
Copyright
© The Authors 2018 

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References

Andreatta, F., Goren, E. Z., Howard, B. and Madapusi Pera, K., Faltings heights of abelian varieties with complex multiplication , Ann. of Math. (2) 187 (2018), 391–531.Google Scholar
Ax, J., On Schanuel’s conjectures , Ann. of Math. (2) 93 (1971), 252–268.Google Scholar
Ax, J., Some topics in differential algebraic geometry. I. Analytic subgroups of algebraic groups , Amer. J. Math. 94 (1972), 1195–1204.Google Scholar
Baily, W. L. Jr. and Borel, A., Compactification of arithmetic quotients of bounded symmetric domains , Ann. of Math. (2) 84 (1966), 442–528.Google Scholar
Bombieri, E., Masser, D. and Zannier, U., Anomalous subvarieties—structure theorems and applications , Int. Math. Res. Not. IMRN 2007 (2007), rnm057.Google Scholar
Borovoi, M., Daw, C. and Ren, J., Conjugation of semisimple subgroups over real number fields of bounded degree, Preprint (2018), arXiv:1802.05894.Google Scholar
Capuano, L., Masser, D., Pila, J. and Zannier, U., Rational points on Grassmannians and unlikely intersections in tori , Bull. Lond. Math. Soc. 48 (2016), 141–154.Google Scholar
Daw, C., A simplified proof of the André–Oort conjecture for products of modular curves , Arch. Math. (Basel) 98 (2012), 433–440.Google Scholar
Daw, C., The André–Oort conjecture via o-minimality , in O-minimality and diophantine geometry, London Mathematical Society Lecture Note Series, vol. 421 (Cambridge University Press, Cambridge, 2015), 129–158.Google Scholar
Daw, C. and Orr, M., Heights of pre-special points of Shimura varieties , Math. Ann. 365 (2016), 1305–1357.Google Scholar
Diamond, F. and Shurman, J., A first course in modular forms, Graduate Texts in Mathematics, vol. 228 (Springer, New York, 2005).Google Scholar
van den Dries, L., Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 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 (1994), 19–56.Google Scholar
Edixhoven, B., Special points on products of modular curves , Duke Math. J. 126 (2005), 325–348.Google Scholar
Edixhoven, B. and Yafaev, A., Subvarieties of Shimura varieties , Ann. of Math. (2) 157 (2003), 621–645.Google Scholar
Edixhoven, S. J., Moonen, B. J. J. and Oort, F., Open problems in algebraic geometry , Bull. Sci. Math. 125 (2001), 1–22.Google Scholar
Gao, Z., About the mixed André–Oort conjecture: reduction to a lower bound for the pure case , C. R. Math. Acad. Sci. Paris 354 (2016), 659–663.Google Scholar
Gao, Z., Towards the André–Oort conjecture for mixed Shimura varieties: the Ax–Lindemann theorem and lower bounds for Galois orbits of special points , J. Reine Angew. Math. 732 (2017), 85–146.Google Scholar
Grauert, H. and Remmert, R., Coherent analytic sheaves, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 265 (Springer, Berlin, 1984).Google Scholar
Habegger, P., Intersecting subvarieties of abelian varieties with algebraic subgroups of complementary dimension , Invent. Math. 176 (2009), 405–447.Google Scholar
Habegger, P. and Pila, J., Some unlikely intersections beyond André–Oort , Compos. Math. 148 (2012), 1–27.Google Scholar
Habegger, P. and Pila, J., O-minimality and certain atypical intersections , Ann. Sci. Éc. Norm. Supér. (4) 49 (2016), 813–858.Google Scholar
Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, New York, 1977).Google Scholar
Klingler, B., Ullmo, E. and Yafaev, A., The hyperbolic Ax-Lindemann-Weierstrass conjecture , Publ. Math. Inst. Hautes Études Sci. 123 (2016), 333–360.Google Scholar
Maurin, G., Courbes algébriques et équations multiplicatives , Math. Ann. 341 (2008), 789–824.Google Scholar
Milne, J. S., Introduction to Shimura varieties , in Harmonic analysis, the trace formula, and Shimura varieties, Clay Mathematics Proceedings, vol. 4 (American Mathematical Society, Providence, RI, 2005), 265–378.Google Scholar
Mok, N., Pila, J. and Tsimerman, J., Ax-Schanuel for Shimura varieties, Preprint (2017),arXiv:1711.02189.Google Scholar
Moonen, B., Linearity properties of Shimura varieties. I , J. Algebraic Geom. 7 (1998), 539–567.Google Scholar
Orr, M., Unlikely intersections involving Hecke correspondences, Preprint (2017),arXiv:1710.04092.Google Scholar
Orr, M., Height bounds and the Siegel property , Algebra Number Theory 12 (2018), 455–478.Google Scholar
Pila, J. and Tsimerman, J., Ax–Lindemann for A g , Ann. of Math. (2) 179 (2014), 659–681.Google Scholar
Pila, J. and Tsimerman, J., Ax–Schanuel for the j-function , Duke Math. J. 165 (2016), 2587–2605.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, vol. 235 (Birkhäuser Boston, Boston, MA, 2005), 251–282.Google Scholar
Pink, R., A common generalization of the conjectures of André-Oort, Manin-Mumford, and Mordell-Lang, Preprint (2005), https://people.math.ethz.ch/∼pink/publications.html.Google Scholar
Rémond, G., Intersection de sous-groupes et de sous-variétés. III , Comment. Math. Helv. 84 (2009), 835–863.Google Scholar
Tsimerman, J., Ax–Schanuel and o-minimality , in O-minimality and diophantine geometry, London Mathematical Society Lecture Note Series, vol. 421, eds Jones, G. O. and Wilkie, A. J. (Cambridge University Press, Cambridge, 2015), 216–221.Google Scholar
Tsimerman, J., The André–Oort conjecture for A g , Ann. of Math. (2) 187 (2018), 379–390.Google Scholar
Ullmo, E., Equidistribution de sous-variétés spéciales. II , J. Reine Angew. Math. 606 (2007), 193–216.Google Scholar
Ullmo, E., Applications du théorème d’Ax–Lindemann hyperbolique , Compos. Math. 150 (2014), 175–190.Google Scholar
Ullmo, E. and Yafaev, A., A characterization of special subvarieties , Mathematika 57 (2011), 263–273.Google Scholar
Ullmo, E. and Yafaev, A., Galois orbits and equidistribution of special subvarieties: towards the André–Oort conjecture , Ann. of Math. (2) 180 (2014), 823–865.Google Scholar
Ullmo, E. and Yafaev, A., Nombre de classes des tores de multiplication complexe et bornes inférieures pour les orbites galoisiennes de points spéciaux , Bull. Soc. Math. France 143 (2015), 197–228.Google Scholar
Ullmo, E. and Yafaev, A., Algebraic flows on Shimura varieties , Manuscripta Math. 155 (2018), 355–367.Google Scholar
Yafaev, A., A conjecture of Yves André’s , Duke Math. J. 132 (2006), 393–407.Google Scholar
Yuan, X. and Zhang, S.-W., On the averaged Colmez conjecture , Ann. of Math. (2) 187 (2018), 533–638.Google Scholar
Zilber, B., Exponential sums equations and the Schanuel conjecture , J. Lond. Math. Soc. (2) 65 (2002), 27–44.Google Scholar