# Comparison between Faltings height and Modular Height

Motivation/Context: In Faltings’ proof of the Mordell conjecture, there is a theorem that establishes a finiteness of abelian varieties with respect to the Faltings height under certain conditions. This statement is inherited from the Northcott property on the modular height. To make the comparison of the two heights above, I am following the exposition in Cornell—Silverman — namely this is the content of Lemma 2 and its corollary in Faltings’ article ‘Finteness Theorems for Abelian Varieties’. The proof introduces an auxiliary stack to show the comparison of the Hodge bundle at a single point on the Siegel moduli space to the Hodge bundle of the universal abelian variety.

Questions: 1. Is the auxiliary stack required in this proof? 2. Can the comparison be made with respect to a different line bundle? In particular, is the role of the universal abelian variety necessary, or can we replace $$\omega_{A/\mathfrak{A}_g}$$ with a different bundle on the moduli stack that does not reference the universal object? 3. Is there a better reference for the fact that $$\omega_{A/\mathfrak{A}_g}$$ is ample on $$A_g/\mathbb{Q}$$ than Bailey-Borel’s article ‘Compactification of arithmetic quotients of bounded symmetric domains’?