Existence and non-existence of rational elliptic curves with prescribed torsion subgroups over quadratic fields

Let $K=\mathbb{Q}(\sqrt{-p})$ be a quadratic field for an odd prime $p$. We show that there exist infinitely many primes $p$ for which no elliptic curve $E\mathrm{/}\mathbb{Q}$ has torsion subgroup $\mathbb{Z}\mathrm{/}2\mathbb{Z}\times \mathbb{Z}\mathrm{/}2N\mathbb{Z}$ over $K$ for $N=5,6$. We also prove that there exist infinitely many primes $p$ for which there are infinitely many elliptic curves $E\mathrm{/}\mathbb{Q}$ with this torsion structure, conditional on the parity conjecture. Using these results, we obtain new torsion classification results over Kummer extensions of cyclotomic fields and over composites of ${\mathbb{Z}}_p$-extensions of number fields, refining and extending our previous work.