Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0616.10025
Carayol, Henri
On the $l$-adic representations associated to Hilbert modular forms. (Sur les représentations $l$-adiques associées aux formes modulaires de Hilbert.) (French)
[J] Ann. Sci. Éc. Norm. Supér. (4) 19, No. 3, 409-468 (1986). ISSN 0012-9593

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 $F$. \par 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 $\sigma\sp{\lambda}(\pi\sb v).$ \par 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. \par 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 ${\Bbb Q}$.
[S. Raghavan]

MSC 2000:: *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

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

Cited in: Zbl 1259.11057 Zbl 1228.11076 Zbl 1183.11023 Zbl 1053.11043 Zbl 0921.11060 Zbl 0877.11034 Zbl 0810.11034 Zbl 0705.11032 Zbl 0629.14017

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Journal Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]