result characterizing such curves, see Theorem \[thm:classification2torsion\]) that allow us to give a complete description of them. The ideas developed here could be adapted to more general situations, but the fact that $K$ has finitely many units appears to play an important role.

The main results of the article are Theorem \[thm:mainthm\] and Corollary \[coro:Szpiro\], which establish Szpiro’s conjecture for “modular” prime power conductor elliptic curves over $K$, assuming a form of level-lowering result (Conjecture \[conj:loweringlevel\]). The general strategy of the proof is as follows: let $E/K$ be a modular elliptic curve (see Conjecture \[conj:modularity\]) of prime power conductor. If $E$ has potentially good reduction, there is a well-known bound for its minimal discriminant ${{\mathcal{D}}}(E)$, so we can focus on the semistable case. If $\ell$ is a rational prime dividing the valuation of ${{\mathcal{D}}}(E)$, by a theorem of Mazur, $E[\ell]$ is a finite flat group scheme. If the residual Galois representation is absolutely irreducible, then the level-lowering conjecture implies the existence of a Bianchi modular form modulo $\ell$ of level $1$ and weight $2$ whose Galois representation matches that of $E$. Since for $\ell$ odd, there are no such forms over the fields in question, we get a contradiction. We 