For a maximal ideal \(\mathfrak m\) of some anemic Hecke \({\overline{{\mathbb Z}}}_l\) -algebra \({\mathbb T}^S_\xi \) of a similitude group of signature \((1,d-1)\) , one can associate a Galois \({\overline{{\mathbb F}}}_l\) -representation \({\overline{\rho }}_{\mathfrak m}\) as well as a Galois \({\mathbb T}_{\xi ,\mathfrak m}^S\) -representation \(\rho _{\mathfrak m}\) . When \(l\ge d\) , and for some split prime \(p \ne l\) , on can also define a monodromy operator \({\overline{N}}_{\mathfrak m}\) as well as \(N_{\widetilde{\mathfrak m}}\) for every minimal prime ideal \(\widetilde{\mathfrak m} \subset \mathfrak m\) , giving rise to partitions \(\underline{{\bar{d}}_{\mathfrak m}}\) and \({\underline{d}}_{\widetilde{\mathfrak m}}\) of d. As with Mazur’s principle for \(GL_2\) , analysing the difference between these partitions, we infer informations about the liftings of \({\overline{\rho }}_{\mathfrak m}\) in characteristic zero known as level lowering problem.
