Let $V\rightarrow A$ be the uniformisation of an abelian variety $A$ over $\mathbb{C}$ by a complex vector space $V$. Identify the period lattice with $\mathbb{Z}^{2g}$ and let $z\in A$ be a torsion point. A pre-image $v\in V$ of $z$ belongs to $\mathbb{Q}^{2g}$ and, if we choose $v\in [0,1)^{2g}$, then the (multiplicative Weil) height of $v$ is bounded by the order of $z$. This observation is essential in the proof of the Manin-Mumford conjecture by Pila and Zannier using o-minimality. Translating their strategy to the André-Oort conjecture for special points of Shimura varieties and, in particular, obtaining analogous bounds for special points, presents a nontrivial problem. In that setting, one considers the uniformisation $X\rightarrow S$ of a Shimura variety $S$ by a hermitian symmetric domain $X$ and, given a special point $z\in S$, one hopes to bound the height of any pre-image $x$ of $z$, belonging to a fixed fundamental set, in terms of the discriminant of the centre of the endomorphism ring of the $\mathbb{Z}$-Hodge structure (or CM abelian variety) associated with $x$. In this talk, we will explain a method for obtaining such bounds.

This is joint work with Martin Orr.