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**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.

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 |

### Keywords:

Hecke algebra; quaternion algebras; 2-dimensional Galois representations; totally real number fields; Hilbert cusp forms; automorphic representation; even degree; congruences; pseudo-representations### References:

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