HILBERT MODULAR FORMS AND p-ADIC HODGE THEORY

We consider 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 place not dividing p is compatible with the local Langlands correspondence [C2]. In this paper, we show that the same is true for the places dividing p, in the sense of p-adic Hodge theory [Fo], as is shown for an elliptic modular form in [Sa]. We also prove that the monodromy-weight conjecture holds such representations. We prove the compatibility by comparing the p-adic and l-adic representations for it is already established for l-adic representation [C2]. More precisely, we prove it by comparing the traces of Galois action and proving the monodromy-weight conjecture. The first task is to construct the Galois representation in purely geometric way in terms of etale cohomology of an analogue of Kuga-Sato variety and algebraic correspondences acting on it. Then we apply the comparison theorem of p-adic Hodge theory [Tj] and weight spectral sequence [RZ], [M] to compute the traces and monodromy operaters in terms of the reduction modulo p. We obtain the required equality between traces by applying Lefschetz trace formula which has the same form for l-adic and for cristalline cohomology. We deduce the monodromy-weight conjecture from the Weil conjecture and a certain vanishing of global sections. The last vanishing result is an analogue of the vanishing of the fixed part (Symk−2TlE)SL2(Zl) for k > 2 for the universal elliptic curve E over a modular curve in positive characteristic. We briefly recall the basic definitions on Hilbert modular forms in Section 1 and an l-adic representation associated to it in Section 2. The main compatibility result, Theorem 1, and the monodromy-weight conjecture, Theorem 2, are stated at the end of Section 2. We recall a cohomological construction of the l-adic representation in Section 3. After introducing Shimura curves in Section 4 and recalling its modular interpretation in Section 5, we give a geometric construction of the l-adic representation in Section 6. We extend the geometric construction to semi-stable