Applications of the hyperbolic Ax-Schanuel conjecture

Daw, C. ORCID: https://orcid.org/0000-0002-2488-6729 and Ren, J. (2018) Applications of the hyperbolic Ax-Schanuel conjecture. Compositio Mathematica, 154 (9). pp. 1843-1888. ISSN 1570-5846 doi: 10.1112/S0010437X1800725X

Abstract/Summary

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.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/72955
Identification Number/DOI	10.1112/S0010437X1800725X
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Publisher	Foundation Compositio Mathematica
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