# Relation between Faltings height and height on moduli space

Let $$E$$ be an elliptic curve over a number field $$K$$. The difference between the semistable Faltings height $$h_F(E)$$ of $$E$$ and the height $$h(j_E)$$ of the $$j$$-invariant of $$E$$ can be bounded in terms of $$h(j_E)$$, see for example Proposition 2.1 in Silverman Heights and Elliptic Curves''.

Are similar results known when we consider a more general abelian variety instead of an elliptic curve, and the height of the corresponding point on some moduli space in place of the height of the $$j$$-invariant? I would be particularly interested in the case of families of abelian varieties (with appropriate extra structure) parametrised by some fixed Shimura curve.