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Applications of the hyperbolic Ax–Schanuel conjecture
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- Journal:
- Compositio Mathematica / Volume 154 / Issue 9 / September 2018
- Published online by Cambridge University Press:
- 13 August 2018, pp. 1843-1888
- Print publication:
- September 2018
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- Article
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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.
