Heavenly elliptic curves over quadratic fields

Abstract An abelian variety A / K is heavenly at $$\ell $$ ℓ if the extension $$K(A[\ell ^\infty ])/K(\mu _{\ell ^{\infty }}\!)$$ K ( A [ ℓ ∞ ] ) / K ( μ ℓ ∞ ) is both pro- $$\ell $$ ℓ and unramified away from $$\ell $$ ℓ . It is known that for a fixed quadratic field K , the number of K -isomorphism classes of heavenly elliptic curves is finite, even running over all primes $$\ell $$ ℓ . We prove a complementary result, that for a fixed prime $$\ell \ge 7$$ ℓ ≥ 7 , there are only finitely many such classes, even running over all quadratic fields. This naturally raises the question of whether to expect a finiteness result when both K and $$\ell $$ ℓ are allowed to vary. We demonstrate similarities in the behavior of heavenly elliptic curves and elliptic curves with complex multiplication, in terms of their Frobenius traces modulo $$\ell $$ ℓ . We determine the complete list of heavenly elliptic curves defined over quadratic fields with complex multiplication and with irrational j -invariant (up to isomorphism). We include various extensions of our results to higher degree fields and higher-dimensional abelian varieties where possible.