Heights of pre-special points of Shimura varieties

Daw, C. ORCID: https://orcid.org/0000-0002-2488-6729 and Orr, M. (2016) Heights of pre-special points of Shimura varieties. Mathematische Annalen, 365 (3-4). pp. 1305-1357. ISSN 0025-5831 doi: 10.1007/s00208-015-1328-3

Abstract/Summary

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 ZZ -Hodge structure. Our bound is the final step needed to complete a proof of the André–Oort conjecture under the conjectural lower bounds for the sizes of Galois orbits of special points, using a strategy of Pila and Zannier.