Heights of pre-special points of Shimura varieties

TL;DR

This paper establishes a polynomial bound on the height of pre-special points in Shimura varieties, advancing the proof of the Andre-Oort conjecture by linking heights to discriminants and Galois orbit sizes.

Contribution

It provides the final key height bound needed to prove the Andre-Oort conjecture under certain conjectural assumptions.

Findings

Height of pre-special points is polynomially bounded by discriminant.

The result completes a crucial step in proving the Andre-Oort conjecture.

Supports the strategy of Pila and Zannier for special points.

Abstract

Let s be a special point on a Shimura variety, and x a pre-image of s in a fixed fundamental set of the associated Hermitian symmetric domain. We prove that the height of x is polynomially bounded with respect to the discriminant of the centre of the endomorphism ring of the corresponding Z-Hodge structure. Our bound is the final step needed to complete a proof of the Andre-Oort conjecture under the conjectural lower bounds for the sizes of Galois orbits of special points, using a strategy of Pila and Zannier.