Conditional Bounds on Heavenly Elliptic Curves over Quadratic Number Fields

<p>Let <em>K</em> be a number field. An elliptic curve<em> E</em> /<em> K </em>is called <strong>heavenly</strong> at a rational prime ℓ if its ℓ-power torsion is constrained in a particular way. In 2008, Rasmussen and Tamagawa conjectured that there are only finitely many <em>K</em>­isomorphism classes of elliptic curves (and, more generally, abelian varieties) defined over <em>K</em> that are heavenly at some prime ℓ. They have since proven the conjecture in some unconditional cases; under the assumption of the Generalized Riemann Hypothesis, they have proven the conjecture in all cases. However, even in the conditional case, existing bounds are impractically large. We explain the new conditional result that for quadratic number fields <em>K</em> of small discriminant, if ℓ > 163, there are no elliptic curves <em>E</em> / <em>K</em> that are heavenly at ℓ.</p>