Heavenly elliptic curves over quadratic fields

TL;DR

This paper investigates the finiteness properties of heavenly elliptic curves over quadratic fields, showing that for a fixed prime, only finitely many such curves exist across all quadratic fields, and classifying those with complex multiplication.

Contribution

It proves finiteness of heavenly elliptic curves over quadratic fields for fixed primes and classifies those with complex multiplication and irrational j-invariant.

Findings

Finiteness of heavenly elliptic curves over quadratic fields for fixed prime $\\ell$

Complete classification of such curves with complex multiplication and irrational j-invariant

Extensions of results to higher degree fields and abelian varieties

Abstract

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. 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.