Automorphic forms and - adic representations 4 Takeshi Saito

In Carayol’s note [4], a geometric construction of the Galois representations associated to Hilbert modular forms and the compatibility with the local Langlands correspondence are discussed. In loc. cit., the compatibility is established in the case = p where the Galois representation is an -adic representation and p is the prime divided by the prime p of the totally real field where the restriction to the decomposition group is considered. The purpose of this note is to sketch the proof of the compatibility in the remaining case p = . In this note, we only discuss the compatibility in the case where the Galois representation is constructed geometrically. Namely, we assume the condition (∗) in Theorem 1 in the text. We only give the main ideas of the proof and refer for the detail to [10]. In other cases, there are alternative arguments using congruences. They cover the cases where the level is prime to p [12] or the residual representation is absolutely irreducible [6] Theorem (4.3). However, the general case still remains open.