Equivalence of conjectures on heavenly elliptic curves

TL;DR

This paper investigates two finiteness conjectures related to heavenly elliptic curves over quadratic fields, demonstrating their equivalence and providing evidence for their validity in the context of number theory.

Contribution

It establishes the equivalence of two conjectures on heavenly elliptic curves, linking their distribution over quadratic fields and isomorphism classes.

Findings

Proved the equivalence of the two conjectures.

Presented evidence supporting the finiteness conjectures.

Connected the distribution of heavenly elliptic curves to their isomorphism classes.

Abstract

Heavenly abelian varieties are abelian varieties defined over number fields that exhibit constrained $\ell$-adic Galois representations for some rational prime $\ell$. 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.