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arXiv · 1703.08967

Applications of the hyperbolic Ax-Schanuel conjecture

Christopher Daw·Jinbo Ren
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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.

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Christopher Daw, Jinbo Ren. 2017-03-27. Applications of the hyperbolic Ax-Schanuel conjecture. https://doi.org/10.1112/s0010437x1800725x

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