Darmon’s large residual image conjecture for GL₂-type abelian varieties

Let $K$ be a totally real field and $L$ a number field. Let $A/L$ be an abelian variety of $\mathrm{GL}_2$-type such that

$$\operatorname{End}_L(A)\otimes \mathbb Q=\operatorname{End}_{\overline{L}}(A)\otimes \mathbb Q\simeq K.$$

For a prime $\mathfrak{p}$ of $K$ above a rational prime $p$, let $\mathbb F_{\mathfrak{p}}$ denote the residue field of $K$ at $\mathfrak{p}$, and consider the mod $p$ representation associated to $A$. Darmon’s large residual image conjecture. There exists a constant $C(L,K)$, depending only on $L$ and $K$, such that whenever the norm of $\mathfrak{p}$ is greater than $C(L,K)$, the image of this representation contains

$$\mathrm{SL}_2(\mathbb F_{\mathfrak{p}}).$$

This conjecture expresses the expectation that Frey varieties without complex multiplication have residual Galois image as large as possible. It is cited in the source as a conjecture of Darmon; its resolution is not specified here.