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On the l-adic representations associated to Hilbert modular forms. (Sur les représentations l-adiques associées aux formes modulaires de Hilbert.) (French) Zbl 0616.10025

Let π= v π v be a cuspidal automorphic representation of GL 2 (F 𝔸 ) where F 𝔸 is the ring of adeles of a totally real algebraic number field F of degree d over , of the same type as representations corresponding to Hilbert modular forms of weight (k 1 ,···,k d ), i.e. whose local components π v for each of the d Archimedean places v of F are essentially square integrable representations of GL 2 () occurring in the induced representation Ind(μ,ν) (under unitary induction) with characters μ, ν of * given by μ(t):=|t| (k-w-1)/2 (sgnt) k , ν(t):=|t| (-k-w+1)/2 for integral k2 and wk(mod2), all depending on v. For d even, π v is taken to be a special or cuspidal representation of GL 2 (F v ), for at least one non-Archimedean place v of F. Let F ¯ be an algebraic closure of F.

The main theorem proved is the following: there exists an algebraic number field E depending on the given π and a strictly compatible system {σ λ } of continuous 2-dimensional E λ -adic representations of Gal(F ¯/F) such that for every non-Archimedean place v of F and λ of E with residue characteristic different from that of v, the restriction σ v λ of σ λ to the local Weil group WF v is equivalent to σ λ (π v )·

What is new here is that the author determines σ v λ for every non-Archimedean place v of F. Moreover, according to Ribet, σ λ turns out to be irreducible and as such, is characterized entirely.

First, for a weaker version of the main theorem, the author gives a (“geometric”) proof and then he uses ”base change for GL(2)” to deduce the main theorem. As a corollary of the main theorem for d=1 and weight k=2, the author shows an affirmative answer to a conjecture on Weil curves over .

Reviewer: S. Raghavan

MSC:
11F70Representation-theoretic methods in automorphic theory
11F67Special values of automorphic L-series, etc
11G25Varieties over finite and local fields
11F41Hilbert modular forms and surfaces