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Hilbert modular forms and \(p\)-adic Hodge theory. (English) Zbl 1259.11060
Summary: For the \(p\)-adic Galois representation associated to a Hilbert modular form, Carayol has shown that, under a certain assumption, its restriction to the local Galois group at a finite place not dividing \(p\) is compatible with the local Langlands correspondence. Under the same assumption, we show that the same is true for the places dividing \(p\), in the sense of \(p\)-adic Hodge theory, as is shown for an elliptic modular form. We also prove that the monodromy-weight conjecture holds for such representations.

MSC:
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
14G35 Modular and Shimura varieties
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