Let $A$ be an abelian variety defined over a number field $K$ of dimension $g \geq 2$, and put $h_F(A)$ for the (stable) Faltings height of $A$. It is well-known from the seminal paper of Faltings that for a given $A$ there are only finitely many abelian varieties $B$ which are isogenous to $A$ over $K$. Note that the Faltings height is not an isogeny invariant but is bounded within the isogeny class. Further, $h_F(A)$ is in fact a $\overline{K}$-isomorphism invariant.

Let $\mathfrak{C}(K,g)$ be the set of isogeny classes of abelian varieties $A$ defined over $K$ of dimension $g$. For a given class $C \in \mathfrak{C}(K,g)$, let $S(C)$ be a set of representatives of $K$-isomorphism classes in the class $C$. By Faltings' theorem, $S(C)$ is finite. Thus the following is well-defined and finite:

$\displaystyle N(C) = \sup_{A,B \in S(C)} |h_F(A) - h_F(B)|.$

Is the quantity $N(C)$ bounded as $C$ varies over $\mathfrak{C}(K,g)$?

A related question: is it expected that $\# S(C)$ is bounded as $C$ varies over $\mathfrak{C}(K,g)$?


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