An abelian variety $A/K$ is heavenly at $\ell$ if the extension $K(A[\ell^{ \infty}])/K(\mu_{ \ell^{ \infty}}\!)$ is both pro-$\ell$ and unramified away from $\ell$. 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 $\ell$. We prove a complementary result, that for a fixed prime $\ell \geq 7$, there are only finitely many such classes, even running over all quadratic fields. We demonstrate similar behavior between heavenly elliptic curves and elliptic curves with complex multiplication, in terms of Frobenius traces modulo $\ell$. We conjecture a stronger finiteness statement based on these new results. Finally, we produce an explicit list of elliptic curves with irrational $j$-invariants that contains all heavenly elliptic curves with complex multiplication defined over quadratic fields, supported by overwhelming computational evidence that every curve on the list is heavenly. We include various extensions of our results to higher degree fields and higher-dimensional abelian varieties where possible.
Categories: math.NT