## 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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