## On Galois representations associated to Hilbert modular forms.(English)Zbl 0705.11031

The author studies the conjecture concerning 2-dimensional Galois representations over totally real number fields $$F$$ attached to Hilbert cusp forms $$f$$ over $$F$$, which are Hecke eigenforms. This conjecture was already established for $$F=\mathbb Q$$ (Eichler, Shimura, Deligne, Deligne–Serre respectively for weight 2, $$\geq 2$$, 1) and for fields of odd degree $$(F:\mathbb Q)$$ by Ohta and, using Shimura curves, by Rogawski-Tunnell. In case of even degree $$(F:\mathbb Q)$$ however, additional assumptions on the automorphic representation $$\pi_ f$$ attached to the form $$f$$ were needed.
The content of this paper is a proof of the conjecture for arbitrary totally real fields $$F$$ of even degree, in case the weights of the forms are $$\geq 2$$. The author proceeds along the lines of Wiles’ approach, who had proved the conjecture for even fields $$F$$, in case the form is ordinary at the relevant prime. The main part of the paper is the proof of congruences between the given form $$f$$ and a suitable newform, for which the conjecture is already established. Then using Wiles’ method of ‘pseudo-representations’ the desired representation is constructed.
Reviewer: N. Klingen

### MSC:

 11F80 Galois representations 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F33 Congruences for modular and $$p$$-adic modular forms 11F70 Representation-theoretic methods; automorphic representations over local and global fields
Full Text:

### References:

 [1] [A] Arthur, J.: The Selberg Trace Formula for Groups ofF-rank One. Ann. Math.100, 326-385 (1974) · Zbl 0257.20033 [2] [BR] Blasius, D., Rogawski, J.: Galois Representations for Hilbert Modular Forms (Preprint) · Zbl 0684.12013 [3] [BL] Brylinski, J.L.: Labesse, J.P.: Cohomologies d’Intersection et FonctionsL de Certaines Variétés de Shimura. Ann. Sci. Ec. Norm. Super. (4)17, 361-412 (1984) · Zbl 0553.12005 [4] [C] Carayol, H.: Sur les Représentationsp-adiques Associées aux Formes Modulaires de Hilbert. Ann. Sci. Ec. Norm. Super. (4)19, 409-468 (1986) · Zbl 0616.10025 [5] [H] Hida, H.: Onp-adic Hecke Algebras forGL 2 over Totally Real Fields. Ann. Math.128, 295-384 (1988) · Zbl 0658.10034 [6] [JL] Jacquet, H., Langlands, R.P.: Automorphic Forms onGL 2. Lect. Notes Math., vol. 114, Berlin Heidelberg New York: Springer 1970 [7] [O] Ohta, M.: On the Zeta Function of an Abelian Scheme over the Shimura Curve. Jpn. J. Math.9, 1-26 (1983) · Zbl 0527.10023 [8] [R] Ribet, K.: Congruence Relations between Modular Forms. Proc. I.C.M. 1983, pp. 503-514 · Zbl 0575.10024 [9] [RT] Rogawski, J.D., Tunnell, J.B.: On ArtinL-functions Associated to Hilbert Modular Forms of Weight One. Invent. Math.74, 1-42 (1983) · Zbl 0523.12009 [10] [Sa] Shimura, G.: The Special Values of the Zeta Functions associated with Hilbert Modular Forms. Duke Math. J.45, 637-679 (1978) · Zbl 0394.10015 [11] [Su] Shimitzu, H.: Theta Series and Modular Forms onGL 2. J. Math. Soc. Jpn24, 638-683 (1973) · Zbl 0241.10016 [12] [W] Wiles, A.: On Ordinary ?-adic Representations Associated to Modular Forms. Invent. Math.94, 529-573 (1988) · 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.