Modular supercuspidal lifts of weight 2

Let F / Q be a totally real number field and N an ideal of its ring of integers of norm N . Let p > max ⁡ { k + 1 , 6 } be a prime totally split in F such that p ∤ N . For every even k ≥ 2 , define the [ F : Q ] -dimensional parallel weight k = ( k , . . . , k ) . Let f ∈ S k ( Γ 0 ( N ) ) be any non CM Hilbert cuspidal Hecke eigenform. Assume that the residual representation ρ ‾ f , P has large image for some prime P over p in the field of definition of f . Under these conditions, we prove that there exists a lift of ρ ‾ f , P associated to a Hilbert modular cuspform g ∈ S 2 ( N p 2 , ϵ ) which is supercuspidal at each prime of F over p . We also give a proof of the corresponding statement for classical Hecke cuspforms. Such statement was already proved by Khare [23] with classical techniques. Finally, using our main result we give a corrigenda for [12] , correctly inserting the micro good dihedral prime in the level.