Hilbert modular forms and local Langlands

The automorphic representation associated to an eigenform will be described, as well as its local factors at the places of F. The statement that the local factor of the automorphic representation determines the local Galois representation (after F-semisimplification) will be explained, and an explicit description will be given in at least the case of a principal series local representation. The main reference used in this text is Saito’s preprint, of which we try to follow the notation. We have Q → F totally real, n := dimQ(F ), I := Hom(F,R) and we fix a bijection {1, . . . , n}−̃→I . 1 Hilbert modular forms, before adèles We begin by recalling some parts of the 2 hours by van der Geer in this seminar. We have H the complex upper half plane, with its action by SL2(R). Then on H we have an action by SL2(R), and SL2(OF ) embeds as a discrete subgroup into SL2(R) using all n disctinct embeddings of F into R. Hence any congruence subgrioup Γ of SL2(OF ) acts on H. Let k = (k1, . . . , kn) be in Z. Then a Hilbert modular form on Γ, of weight k, is a holomorphic function: f : H −→ C, such that, for all γ = ( a b c d ) in Γ and all z in H we have: f(γz) = (cz + d)f(z), where (cz + d) = (c1z1 + d1)1 · · · (cnzn + dn) . If F = Q, one asks moreover f to be “holomorphic at the cusps”. Equivalently: f gives a section of some holomorphic line bundle on a compactification Γ\Hn (and compactifying is not necessary for n > 1).