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Level lowering for modular \(\text{mod }\ell\) representations over totally real fields. (English) Zbl 0978.11020

The author continues the study of part of the analogue of Serre’s conjecture for mod \(\ell\) Galois representations for totally real fields. More precisely, one knows, through results of Carayol and Taylor, that to any Hilbert cuspidal eigenform over a totally real field \(F\), one can attach a compatible system of \(\lambda\)-adic representations of the corresponding absolute Galois group. One may ask if a given \(\lambda\)-adic or modulo \(\ell\) representation is attached by this process to a Hilbert modular form, and, if so, what weights and levels this form can have. He proves some analogues of results known in the case \(F= \mathbb Q\).

MSC:

11F33 Congruences for modular and \(p\)-adic modular forms
11F80 Galois representations
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11G18 Arithmetic aspects of modular and Shimura varieties
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