## 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 $$\pi =\otimes_{v}\pi_ v$$ be a cuspidal automorphic representation of $$\text{GL}_ 2(F_{{\mathbb A}})$$ where $$F_{{\mathbb A}}$$ is the ring of adeles of a totally real algebraic number field $$F$$ of degree $$d$$ over $${\mathbb Q}$$, of the same type as representations corresponding to Hilbert modular forms of weight $$(k_ 1,...,k_ d)$$, i.e. whose local components $$\pi_ v$$ for each of the $$d$$ Archimedean places $$v$$ of $$F$$ are essentially square integrable representations of $$\text{GL}_ 2({\mathbb R})$$ occurring in the induced representation $$\text{Ind}(\mu,\nu)$$ (under unitary induction) with characters $$\mu$$, $$\nu$$ of $${\mathbb R}^*$$ given by $$\mu (t):=| t|^{(k-w- 1)/2}(\text{sgn}\, t)^ k$$, $$\nu(t):=| t|^{(-k-w+1)/2}$$ for integral $$k\geq 2$$ and $$w\equiv k\pmod 2$$, all depending on $$v$$. For $$d$$ even, $$\pi_ v$$ is taken to be a special or cuspidal representation of $$\text{GL}_ 2(F_ v)$$, 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 $$\{\sigma^{\lambda}\}$$ of continuous 2-dimensional $$E_{\lambda}$$-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 $$\sigma_ v^{\lambda}$$ of $$\sigma^{\lambda}$$ to the local Weil group $$WF_ v$$ is equivalent to $$\sigma^{\lambda}(\pi_ v).$$
What is new here is that the author determines $$\sigma_ v^{\lambda}$$ for every non-Archimedean place $$v$$ of $$F$$. Moreover, according to Ribet, $$\sigma^{\lambda}$$ 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 $${\mathbb Q}$$.
Reviewer: S. Raghavan

### MSC:

 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11G25 Varieties over finite and local fields 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
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