Modular supercuspidal lifts of weight $2$

Abstract: Let $F/\mathbb{Q}$ be any totally real number field and $\frak{N}$ an ideal of its ring of integers of norm $N$ and define, for every even $n$, the $[F:\mathbb{Q}]$-dimensional multiweight $\textbf{n}=(n,...,n)$. We prove that for a non CM Hilbert cuspidal Hecke eigenform for $F$, say $f\in S_{\textbf{k}}(\Gamma_0(\frak{N}))$ with $k>2$ even, and a prime $p>\max{k+1,6}$ totally split in $F$ such that $p\nmid N$ and such that the residual mod $p$ representation $\overline{\rho}f$ satisfies that $\mathrm{SL}_2(\mathbb{F}_p)\subseteq \mathrm{Im}(\overline{\rho}_f)$, there exists a lift $\rho_g$ associated to a Hilbert modular cuspform for $F$, say $g\in S{\textbf{2}}(\frak{N}p^2,\epsilon)$ for some Nebentypus character $\epsilon$ which is supercuspidal at each prime of $F$ over $p$. We also observe that our techniques provide an alternative proof to the corresponding statement for classical Hecke cuspforms already proved by Khare \cite{khare} with classical techniques. Finally, we take the opportunity to include a corrigenda for \cite{dieulefait} which follows from our main result, which provides a congruence that puts the micro good dihedral prime in the level.