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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 $\pi ={\otimes }_{v}{\pi }_{v}$ be a cuspidal automorphic representation of ${\text{GL}}_{2}\left({F}_{𝔸}\right)$ where ${F}_{𝔸}$ is the ring of adeles of a totally real algebraic number field $F$ of degree $d$ over $ℚ$, of the same type as representations corresponding to Hilbert modular forms of weight $\left({k}_{1},···,{k}_{d}\right)$, i.e. whose local components ${\pi }_{v}$ for each of the $d$ Archimedean places $v$ of $F$ are essentially square integrable representations of ${\text{GL}}_{2}\left(ℝ\right)$ occurring in the induced representation $\text{Ind}\left(\mu ,\nu \right)$ (under unitary induction) with characters $\mu$, $\nu$ of ${ℝ}^{*}$ given by $\mu \left(t\right):={|t|}^{\left(k-w-1\right)/2}{\left(\text{sgn}\phantom{\rule{0.166667em}{0ex}}t\right)}^{k}$, $\nu \left(t\right):={|t|}^{\left(-k-w+1\right)/2}$ for integral $k\ge 2$ and $w\equiv k\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}2\right)$, all depending on $v$. For $d$ even, ${\pi }_{v}$ is taken to be a special or cuspidal representation of ${\text{GL}}_{2}\left({F}_{v}\right)$, for at least one non-Archimedean place $v$ of $F$. Let $\overline{F}$ 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 $\left\{{\sigma }^{\lambda }\right\}$ of continuous 2-dimensional ${E}_{\lambda }$-adic representations of $\text{Gal}\left(\overline{F}/F\right)$ 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 $W{F}_{v}$ is equivalent to ${\sigma }^{\lambda }\left({\pi }_{v}\right)·$

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 $\text{GL}\left(2\right)$” 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 $ℚ$.

Reviewer: S. Raghavan

##### MSC:
 11F70 Representation-theoretic methods in automorphic theory 11F67 Special values of automorphic $L$-series, etc 11G25 Varieties over finite and local fields 11F41 Hilbert modular forms and surfaces