Applications of the hyperbolic Ax–Schanuel conjecture

By Christopher Daw, Jinbo Ren. Compositio Mathematica, 2017

53 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6b0970ef0cfe48087f043c099b39e28fba519e6d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

1. Bruno Klingler
2. E. Ullmo
3. G. Baldi
4. Ziyang Gao

In Geometry and Topology

1. Bruno Klingler
2. E. Ullmo
3. G. Baldi
4. Jacob Tsimerman
5. Jonathan Pila
6. Ziyang Gao
7. A. Yafaev

Citations in the corpus

• On the distribution of the Hodge locus (2021, methodology): “Starting from Pila and Zannier [50], a link between functional transcendence and the Zilber–Pink conjecture has been observed in [60, 31, 19], at least for subvarieties of abelian and pure Shimura varieties.”
• Ax-Schanuel for Shimura varieties (2017, unknown): “One such application has been given by Daw-Ren [7].”