form in Conjecture \[conj:loweringlevel\] is not expected to be global.

Modular elliptic curves of prime power conductor
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Recall the following result due to Serre and Mestre-Oesterlé (see [@Mestre]).

Let $E/\Q$ be a modular elliptic curve of prime conductor $p$. If $p >37$ then ${{\mathcal{D}}}(E) = \pm p$ up to $2$-isogenies (i.e., either ${{\mathcal{D}}}(E) = \pm p$, or there exists a curve $2$-isogenous to $E$ over $\Q$ with discriminant $\pm p$). \[thm:MO\]

A key ingredient in the proof is the following result of Mazur.

Let $K$ be a number field, let $E/K$ be a semistable elliptic curve with discriminant ${{\mathcal{D}}}(E)$, and let $\ell$ be a prime number. Then $E[\ell]$ is a finite flat group scheme if and only if $\ell \mid v_{\id{q}}({{\mathcal{D}}}(E))$ for all primes $\id{q}$ in ${{\mathcal{O}}}_K$, i.e. if the ideal $({{\mathcal{D}}}(E))$ is an $\ell$-th power. \[thm:Mazur\]

See [@Mazur2 Proposition 9.1].

If $E/K$ is an elliptic curve of prime conductor $\id{p}$, and $\ell$ is a prime dividing the valuation $v_{\id{p}}({{\mathcal{D}}}(E))$ of ${{\mathcal{D}}}(E)$, then $E[\ell]$ is a finite flat group scheme over $\operatorname{Spec}({{\mathcal{O}}}_K)$ by Mazur’s theorem. To apply a level-lowering result such as Conjecture \[conj:loweringlevel\], one also needs a *big image* hypothesis, i.e. that the residual representation at $\ell$ is 