Let $K$ be a number field. Suppose we order elliptic curves over $K$ by naive height. What is the natural density of elliptic curves without complex multiplication?
More generally, suppose we order $g$-dimensional abelian varieties over $K$ by Faltings height. What is the natural density of such varieties without complex multiplication?
Best Answer
The natural density of elliptic curves with complex multiplication is 0 (say we order by the coefficients A, B in y^2 = x^3 + Ax + B). This follows by Proposition 5 of http://arxiv.org/abs/0804.2166 combined with Hilbert irreducibility:
Since the family of elliptic curves given above has surjective mod-\ell Galois representation for all \ell, it follows from Hilbert irreducibility that a density 1 subset of its members have surjective mod \ell image. Then, by Proposition 5, these members have trivial endomorphism ring. This same proof works for rational families of arbitrary genus with surjective mod \ell Galois representation.
Note that the assumption on rationality is important to apply Hilbert irreducibility, so the same proof will not go through for the moduli space of Abelian varieties in genus more than 7, as the moduli space is not rational (or even unirational).