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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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