Level lowering: a Mazur principle in higher dimension

TL;DR

This paper extends the Mazur principle to higher-dimensional similitude groups, analyzing Galois representations and monodromy operators to understand level lowering in the context of Hecke algebras.

Contribution

It introduces a higher-dimensional analogue of Mazur's principle, relating monodromy partitions to Galois representation liftings in a new setting.

Findings

Established a relation between monodromy partitions and Galois liftings.

Extended Mazur's level lowering principle to higher dimensions.

Provided new tools for analyzing Galois representations in automorphic forms.

Abstract

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 liftings of $\overline \rho_{\mathfrak m}$ in characteristic zero known as level lowering problem.