*(French)*Zbl 0616.10025

Let $\pi ={\otimes}_{v}{\pi}_{v}$ be a cuspidal automorphic representation of ${\text{GL}}_{2}\left({F}_{\mathbb{A}}\right)$ 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 $({k}_{1},\xb7\xb7\xb7,{k}_{d})$, i.e. whose local components ${\pi}_{v}$ for each of the $d$ Archimedean places $v$ of $F$ are essentially square integrable representations of ${\text{GL}}_{2}\left(\mathbb{R}\right)$ occurring in the induced representation $\text{Ind}(\mu ,\nu )$ (under unitary induction) with characters $\mu $, $\nu $ of ${\mathbb{R}}^{*}$ given by $\mu \left(t\right):={\left|t\right|}^{(k-w-1)/2}{\left(\text{sgn}\phantom{\rule{0.166667em}{0ex}}t\right)}^{k}$, $\nu \left(t\right):={\left|t\right|}^{(-k-w+1)/2}$ for integral $k\ge 2$ and $w\equiv k\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}2)$, 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 $\text{Gal}(\overline{F}/F)$ 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 $W{F}_{v}$ is equivalent to ${\sigma}^{\lambda}\left({\pi}_{v}\right)\xb7$

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 $\mathbb{Q}$.

##### 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 |