Level lowering: a Mazur principle in higher dimension
Résumé
For a maximal ideal $\mathfrak m$ of some anemic Hecke 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 $\rho_{\mathfrak m}$.For $l\geq d$, on can also define a monodromy operator
$\overline N_{\mathfrak m}$ as well as
$N_{\widetilde{\mathfrak m}}$ for every 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 set of prime ideals $\widetilde{\mathfrak m} \subset \mathfrak m$, i.e.
the liftings of $\overline \rho_{\mathfrak m}$ in characteristic zero known
as level lowering problem.
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