Applications of the hyperbolic Ax-Schanuel conjecture

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.

46 pages; published in Compositio Mathematica; minor revision

References in corpus (1)

• Height bounds and the Siegel property

Cited by in corpus (13)

• Generic rank of Betti map and unlikely intersections
• Quantitative reduction theory and unlikely intersections
• Ax-Schanuel and strong minimality for the $j$-function
• Weak Modular Zilber-Pink with Derivatives
• Differential Existential Closedness for the $j$-function
• Some cases of the Zilber-Pink conjecture for curves in ${\mathcal{A}}_g$
• Effective estimates for the degrees of maximal special subvarieties
• Attractors are not algebraic
• Lattices with skew-Hermitian forms over division algebras and unlikely intersections
• Sets of Special Subvarieties of Bounded Degree
• Galois conjugates of special points and special subvarieties in Shimura varieties
• Rich representations and superrigidity
• Likely intersections