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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 $\pi =\otimes\sb{v}\pi\sb v$ be a cuspidal automorphic representation of $\text{GL}\sb 2(F\sb{{\Bbb A}})$ where $F\sb{{\Bbb A}}$ is the ring of adeles of a totally real algebraic number field $F$ of degree $d$ over ${\Bbb Q}$, of the same type as representations corresponding to Hilbert modular forms of weight $(k\sb 1,...,k\sb d)$, i.e. whose local components $\pi\sb v$ for each of the $d$ Archimedean places $v$ of $F$ are essentially square integrable representations of $\text{GL}\sb 2({\Bbb R})$ occurring in the induced representation $\text{Ind}(\mu,\nu)$ (under unitary induction) with characters $\mu$, $\nu$ of ${\Bbb R}\sp*$ given by $\mu (t):=\vert t\vert\sp{(k-w- 1)/2}(\text{sgn}\, t)\sp k$, $\nu(t):=\vert t\vert\sp{(-k-w+1)/2}$ for integral $k\ge 2$ and $w\equiv k\pmod 2$, all depending on $v$. For $d$ even, $\pi\sb v$ is taken to be a special or cuspidal representation of $\text{GL}\sb 2(F\sb v)$, for at least one non-Archimedean place $v$ of $F$. Let $\bar F$ be an algebraic closure of $F$. The main theorem proved is the following: there exists an algebraic number field $E$ depending on the given $\pi$ and a strictly compatible system $\{\sigma\sp{\lambda}\}$ of continuous 2-dimensional $E\sb{\lambda}$-adic representations of $\text{Gal}(\bar F/F)$ such that for every non-Archimedean place $v$ of $F$ and $\lambda$ of $E$ with residue characteristic different from that of $v$, the restriction $\sigma\sb v\sp{\lambda}$ of $\sigma\sp{\lambda}$ to the local Weil group $WF\sb v$ is equivalent to $\sigma\sp{\lambda}(\pi\sb v).$ What is new here is that the author determines $\sigma\sb v\sp{\lambda}$ for every non-Archimedean place $v$ of $F$. Moreover, according to Ribet, $\sigma\sp{\lambda}$ 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 $\text{GL}(2)$” 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 ${\Bbb Q}$.
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
WorldCat.org
Full Text: Numdam EuDML
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