×

zbMATH — the first resource for mathematics

Galois representations for Hilbert forms. (English) Zbl 0684.12013
Let F be a totally real number field and let \({\mathbb{A}}_ F\) be its adele ring. Let \(\pi\) be a cuspidal automorphic representation of \(GL_ 2({\mathbb{A}}_ F)\) such that for each infinite place v, \(\pi_ v\) is a discrete series representation of weight \(k_ v\) and central character \(t\to t^{-w}\), where w is an integer independent of v. Then \(\pi\) corresponds to a holomorphic Hilbert modular new form of weight \((k_ v).\)
In this paper the authors prove the following: Suppose all \(k_ v\equiv w mod 2.\) Then there exists a number field \(T\subset {\mathbb{C}}\) and a collection \(\rho (\pi)=\{\rho_{\lambda}\}\), where for each \(\ell\)-adic completion \(T_{\lambda}\) of T, \(\rho_{\lambda}\) is a continuous representation of Gal\((\bar F/F)\) in \(GL_ 2(T_{\lambda})\) such that the L-functions \(L_ v(s,\rho_{\lambda})=L_ v(s,\pi)\) for all finite places v prime to \(\ell\) of F at which \(\pi_ v\) is unramified. Furthermore, the system \(\rho\) (\(\pi)\) is motivic.
The main part of the proof is to construct, for each quadratic CM extension E/F, the \(\ell\)-adic representations associated to the L- function of the base change to \(GL_ 2({\mathbb{A}}_ E)\) of certain cuspidal representations of the quasi-split unitary group in two variables relative to E/F in the étale cohomology of local systems on a Shimura surface associated to a certain unitary group in three variables.
Recently, R. Taylor [in Automorphic forms, Shimura varieties, and L-functions II, Academic Press, Boston, 323-336 (1990)] obtained a similar result, but the construction is quite different.
Reviewer: T.Takeuchi

MSC:
11R56 Adèle rings and groups
11R42 Zeta functions and \(L\)-functions of number fields
11R39 Langlands-Weil conjectures, nonabelian class field theory
14G25 Global ground fields in algebraic geometry
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F70 Representation-theoretic methods; automorphic representations over local and global fields
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Don Blasius and Dinakar Ramakrishnan, Maass forms and Galois representations, Galois groups over \? (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 33 – 77. · Zbl 0699.10043
[2] D. Blasius and J. Rogawski, Tate cycles and quotients of the two-ball, to appear in [M]. · Zbl 0828.14012
[3] Henri Carayol, Sur les représentations \?-adiques associées aux formes modulaires de Hilbert, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 409 – 468 (French). · Zbl 0616.10025
[4] P. Deligne, Formes modulaires et représentations l-adiques, Séminaire Bourbaki 355 (Février 1969), SLN 179, Springer-Verlag, New York, pp. 139-172.
[5] Gerd Faltings, \?-adic Hodge theory, J. Amer. Math. Soc. 1 (1988), no. 1, 255 – 299. · Zbl 0764.14012
[6] Proceedings of a Conference on Shimura Varieties, Centre de recherches mathématiques, Université de Montréal, 1988 (in preparation).
[7] Masami Ohta, On the zeta function of an abelian scheme over the Shimura curve, Japan. J. Math. (N.S.) 9 (1983), no. 1, 1 – 25. · Zbl 0527.10023
[8] Jonathan D. Rogawski, Automorphic representations of unitary groups in three variables, Annals of Mathematics Studies, vol. 123, Princeton University Press, Princeton, NJ, 1990. · Zbl 0724.11031
[9] J. Rogawski, article in [M].
[10] J. D. Rogawski and J. B. Tunnell, On Artin \?-functions associated to Hilbert modular forms of weight one, Invent. Math. 74 (1983), no. 1, 1 – 42. · Zbl 0523.12009
[11] Richard Taylor, On Galois representations associated to Hilbert modular forms. II, Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993) Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, pp. 185 – 191. · Zbl 0836.11017
[12] A. Wiles, On ordinary \lambda -adic representations associated to modular forms, Preprint. · Zbl 0664.10013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.