The Faltings height is a useful invariant for addressing questions in

arithmetic geometry. In his celebrated proof of the Mordell and

Shafarevich conjectures, Faltings shows the Faltings height satisfies a

certain Northcott property, which allows him to deduce his finiteness

statements. In this work we prove a new Northcott property for the

Faltings height. Namely we show, assuming the Colmez Conjecture and the

Artin Conjecture, that there are finitely many CM abelian varieties of a

fixed dimension which have bounded Faltings height. The technique

developed uses new tools from integral p-adic Hodge theory to study the

variation of Faltings height within an isogeny class of CM abelian

varieties. In special cases, we are able to use these techniques to

moreover develop new Colmez-type formulas for the Faltings height.

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