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Une remarque sur les représentations locales \(p\)-adiques et les congruences entre formes modulaires de Hilbert. (A remark on local \(p\)-adic Galois representations and congruences between Hilbert modular forms.). (French) Zbl 0933.11028

Let \(F\) be a totally real number field of degree \(d\) and \(f\) be a Hilbert modular form for some congruence subgroup of \(\text{SL}(2,{\mathcal O}_F)\). If every entry of the weight vector \(k=(k_1,\dots,k_d)\) of \(f\) is at least 2, one knows, due to Carayol and Taylor, how to attach to \(f\) a compatible system of \(p\)-adic Galois representations. In [Invent. Math. 114, 55-87 (1993; Zbl 0829.11028)] D. Blasius and J. D. Rogawski have given another construction for the representations, which simultaneously showed that the representations are crystalline for sufficiently large \(p\). The present paper gives an alternative proof of this crystallinity assertion which is close to Taylor’s original construction in that it uses congruences between Hilbert modular forms.
It would be nice to know whether the proof carries over to the case that not all \(k_i\) are at least 2 but may be 1 as well. The existence of a representation in this case is a result of F. Jarvis [J. Reine Angew. Math. 491, 199-216 (1997; Zbl 0914.11025)].

MSC:

11F80 Galois representations
11F33 Congruences for modular and \(p\)-adic modular forms
11F85 \(p\)-adic theory, local fields
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
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References:

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