Level lowering: a Mazur principle in higher dimension

Abstract

For a maximal ideal $\mathfrak{m}$ of some anemic Hecke algebra ${\mathbb{T}}_{\xi }^S$ 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 $l\ge d$, on can also define a monodromy operator ${\overline{N}}_{\mathfrak{m}}$ as well as $N_{\tilde{\mathfrak{m}}}$ for every prime ideal $\tilde{\mathfrak{m}}\subset \mathfrak{m}$, giving rise to partitions $\underline{{\bar{d}}_{\mathfrak{m}}}$ and ${\underline{d}}_{\tilde{\mathfrak{m}}}$ of $d$. As with Mazur's principle for $G{L}_2$, 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.

arXiv comment40