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Mixed Ax–Schanuel for the universal abelian varieties and some applications

Part of: Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry

Published online by Cambridge University Press:  11 December 2020

Ziyang Gao
Ziyang Gao*
Affiliation:
CNRS, IMJ-PRG, 4 place Jussieu, 75005Paris, Franceziyang.gao@imj-prg.fr Department of Mathematics, Princeton University, Princeton, NJ08544, USA
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Abstract

In this paper we prove the mixed Ax–Schanuel theorem for the universal abelian varieties (more generally any mixed Shimura variety of Kuga type), and give some simple applications. In particular, we present an application for studying the generic rank of the Betti map.

Keywords

Universal abelian varietyAx–Schanuelmixed Shimura varieties

MSC classification

Primary: 11G10: Abelian varieties of dimension $> 1$ 11G18: Arithmetic aspects of modular and Shimura varieties 14G35: Modular and Shimura varieties
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 11 , November 2020 , pp. 2263 - 2297
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Copyright
© The Author(s) 2020

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References

André, Y., Corvaja, P. and Zannier, U., The Betti map associated to a section of an abelian scheme (with an appendix by Z. Gao), Invent. Math. 222 (2020), 161–202.10.1007/s00222-020-00963-wCrossRefGoogle Scholar
André, Y., Mumford–Tate groups of mixed Hodge structures and the theorem of the fixed part, Compos. Math. 82 (1992), 1–24.Google Scholar
Ax, J., On Schanuel's conjectures, Ann. of Math. (2) 93 (1971), 252–268.10.2307/1970774CrossRefGoogle Scholar
Ax, J., Some topics in differential algebraic geometry I: analytic subgroups of algebraic groups, Amer. J. Math. 94 (1972), 1195–1204.CrossRefGoogle Scholar
Bakker, B. and Tsimerman, J., The Ax-Schanuel conjecture for variations of Hodge structures, Invent. Math. 217 (2019), 77–94.10.1007/s00222-019-00863-8CrossRefGoogle Scholar
Bogomolov, F., Points of finite order on an abelian variety, Izv. Math. 17 (1981), 55–72.10.1070/IM1981v017n01ABEH001329CrossRefGoogle Scholar
Daw, C. and Ren, J., Applications of the hyperbolic Ax–Schanuel conjecture, Compos. Math. 154 (2018), 1843–1888.10.1112/S0010437X1800725XCrossRefGoogle Scholar
Gao, Z., A special point problem of André-Pink-Zannier in the universal family of abelian varieties, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017), 231–266.Google Scholar
Gao, Z., Towards the André-Oort conjecture for mixed Shimura varieties: the Ax-Lindemann-Weierstrass theorem and lower bounds for Galois orbits of special points, J. Reine Angew. Math. 732 (2017), 85–146.CrossRefGoogle Scholar
Genestier, A. and Ngô, B. C., Lecture on Shimura varieties, 2006, https://www.math.uchicago.edu/ngo/Shimura.pdf.Google Scholar
Habegger, P. and Pila, J., O-minimality and certain atypical intersections, Ann. Sci. cole Norm. Sup. 4 (2016), 813–858.10.24033/asens.2296CrossRefGoogle Scholar
Hwang, J. and To, W., Volumes of complex analytic subvarieties of Hermitian symmetric spaces, Amer. J. Math. 124 (2002), 1221–1246.10.1353/ajm.2002.0038CrossRefGoogle Scholar
Klingler, B., Ullmo, E. and Yafaev, A., The hyperbolic Ax-Lindemann-Weierstrass conjecture, Publ. Math. Inst. Hautes Études Sci. 123 (2016), 333–360.CrossRefGoogle Scholar
Klingler, B., Ullmo, E. and Yafaev, A., Bi-algebraic geometry and the André-Oort conjecture, in Algebraic geometry: Proc. Symp. on pure mathematics, Salt Lake City 2015, Part 2 (Utah, 2015) (American Mathematical Society, 2018), 319–360.CrossRefGoogle Scholar
Milne, J., Canonical models of (mixed) Shimura varieties and automorphic vector bundles, in Proc. conf. automorphic forms, Shimura varieties, and L-functions. Vol. I, University of Michigan, Ann Arbor, Michigan, July 6–16, 1988 (Academic Press, 1990).Google Scholar
Mok, N., Pila, J. and Tsimerman, J., Ax-Schanuel for Shimura varieties, Ann. of Math. (2) 189 (2019), 945–978.10.4007/annals.2019.189.3.7CrossRefGoogle Scholar
Pila, J., O-minimality and the André-Oort conjecture for ${C}^n$, Ann. of Math. (2) 173 (2011), 1779–1840.10.4007/annals.2011.173.3.11CrossRefGoogle Scholar
Pink, R., Arithmetical compactification of mixed Shimura varieties, PhD thesis, Bonner Mathematische Schriften (1989).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. 253 (Birkhäuser, 2005), 251–282.CrossRefGoogle Scholar
Platonov, V. and Rapinchuk, A., Algebraic groups and number theory (Academic Press, 1994).Google Scholar
Peterzil, Y. and Starchenko, S., Complex analytic geometry and analytic-geometric categories, J. Reine Angew. Math. (Crelle) 626 (2009), 39–74.Google Scholar
Peterzil, Y. and Starchenko, S., Definability of restricted theta functions and families of abelian varieties, Duke Math. J. 162 (2013), 731–765.CrossRefGoogle Scholar
Pila, J. and Tsimerman, J., Ax-Schanuel for the $j$-function, Duke Math. J. 165 (2014), 2587–2605.CrossRefGoogle Scholar
Tsimerman, J., Ax-Schanuel and o-minimality, in O-minimality and Diophantine geometry, London Mathematical Society Lecture Note Series, vol. 421 (Cambridge University Press, 2015).Google Scholar
Ullmo, E., Applications du théorème de Ax–Lindemann hyperbolique, Compos. Math. 150 (2014), 175–190.10.1112/S0010437X13007446CrossRefGoogle Scholar
Ullmo, E. and Yafaev, A., A characterisation of special subvarieties, Mathematika 57 (2011), 263–273.10.1112/S0025579311001628CrossRefGoogle Scholar
Wildeshaus, J., The canonical construction of mixed sheaves on mixed Shimura varieties, in Realizations of polylogarithms, Lecture Notes in Mathematics, vol. 1650 (Springer, 1997), 77–140.10.1007/BFb0093054CrossRefGoogle Scholar