# Counting elliptic curves over a number field by their Faltings height

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.