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

Carayol, Henri

Ann. Sci. Éc. Norm. Supér. (4) 19, No. 3, 409-468 (1986).

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

Keywords:

\(\ell\)-adic representations; cuspidal automorphic representation; Hilbert modular forms; base change for GL(2); Weil curves over Q

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