In this paper, Hortsch gives an asymptotic formula for the number of elliptic curves over $\mathbb{Q}$, given by their minimal Weierstrass model, of bounded Faltings' height.
In general, is it possible to count semi-stable elliptic curves (i.e., give a precise asymptotic formula) defined over a number field $K$ of bounded Faltings' height? The semi-stability assumption is to eliminate the ambiguity between Faltings' height and stable Faltings height.
