Maps between Jacobians of Shimura curves and congruence kernels

Let p ≥ 5 be a prime, and℘ a place ofQ above it.We denote byGQ the absolute Galois group Gal(Q/Q). For a finite field k of characteristic p, an absolutely irreducible representation ρ : GQ → GL2(k) is said to arise from a newform of weight k and level a positive integer M if ρ is isomorphic to the reduction mod ℘ of the ℘-adic representation associated to a newform of weight k and level M . Note that then it is known that ρ also arises from a newform of weight 2 and level dividingMp2. Thus we assume without loss of generality that ρ arises from a newform of weight 2 and level a positive integer N . The question of “raising” the level of ρ, i.e., determining when a ρ arising from a newform of weight 2 and level N as above also arises from a newform of weight 2 and levelN ′, whereN dividesN ′, has been widely studied by numerous mathematicians (cf. [10,8,5,2]). In the first paper that considers this kind of question [10], as well as in most of the papers dealing with similar questions (e.g., [2]), the cases considered correspond to N ′/N being coprime to p. Some discussion for the case where