Heavenly elliptic curves over quadratic fields

Abstract

An abelian variety $A/K$ is heavenly at $\ell$ if the extension $K(A[\ell^\infty])/K(μ_{\ell^{\infty}}\!)$ 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. This naturally raises the question of whether to expect a finiteness result when both $K$ and $\ell$ 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 $\ell$. 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.