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Preprints, Working Papers, ... Year : 2023

Level lowering: a Mazur principle in higher dimension

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 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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Dates and versions

hal-02267548 , version 1 (19-08-2019)
hal-02267548 , version 2 (02-07-2021)
hal-02267548 , version 3 (09-01-2022)
hal-02267548 , version 4 (04-12-2022)
hal-02267548 , version 5 (11-10-2023)

Identifiers

  • HAL Id : hal-02267548 , version 5

Cite

Pascal Boyer. Level lowering: a Mazur principle in higher dimension. 2023. ⟨hal-02267548v5⟩
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