Equivalence of conjectures on heavenly elliptic curves

Heavenly abelian varieties are abelian varieties defined over number fields that exhibit constrained $\mathrm{\ell }$-adic Galois representations for some rational prime $\mathrm{\ell }$. At the ICMS Workshop held in November 2024, we presented evidence for two finiteness conjectures around the distribution of heavenly elliptic curves over quadratic number fields, one in terms of isomorphism classes of curves, and a second in terms of quadratic fields which admit heavenly elliptic curves. We prove these two conjectures are equivalent.