Decomposition theorems for Hilbert modular forms.

*(English)*Zbl 1278.11054Summary: Let \(\mathcal{S}_k^+(\mathcal{N},\Phi)\) denote the space generated by Hilbert modular newforms (over a fixed totally real field \(K\)) of weight \(k\), level \(\mathcal{N}\) and Hecke character \(\Phi\). In this paper we examine the behavior of \(\mathcal{S}_k^+(\mathcal{N},\Phi)\) under twists (by a Hecke character). We show how this space may be decomposed into a direct sum of twists of other spaces of newforms. This sheds light on the behavior of a newform under a character twist: the exact level of the twist of a newform, when such a twist is itself a newform, and when a newform may be realized as the twist of a primitive newform. In certain cases it is shown that the entire space \(\mathcal{S}_k^+(\mathcal{N},\Phi)\) can be represented as a direct sum of twists of primitive nebenspaces. This adds perspective to the Jacquet-Langlands correspondence, which characterizes those elements of \(\mathcal{S}_k^+(\mathcal{N},\Phi)\) not representable as theta series arising from a quaternion algebra as being precisely those forms which are twists of primitive nebenforms. It follows that in these cases no newforms arise from a quaternion algebra. These results were proven for elliptic modular forms by Hijikata, Pizer and Shemanske by employing the Eichler-Selberg trace formula.

##### MSC:

11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |

11F11 | Holomorphic modular forms of integral weight |

##### References:

[1] | P. Clark, There are genus one curves of every index over every number field. J. Reine Angew. Math. 594 (2006), 201-206. · Zbl 1097.14024 |

[2] | C.D. Gonzalez-Aviles, Brauer groups and Tate-Shafarevich groups , J. Math. Sci. Univ. Tokyo 10 (2003), 391-419. · Zbl 1029.11026 |

[3] | S. Lichtenbaum, The period-index problem for elliptic curves , Amer. J. of Math. 90 (1968), 1209-1223. · Zbl 0187.18602 |

[4] | S. Lichtenbaum, Duality theorems for curves over p-adic fields , Inventiones Math. 7 (1969), 120-136. · Zbl 0186.26402 |

[5] | J. Milne, Comparison of the Brauer group with the Tate-Shafarevich group , J. Fac. Science, Univ. Tokyo, Sec. IA V. Scharaschkin, The Brauer-Manin obstruction for curves . · Zbl 0503.14010 |

[6] | J. Silverman, The arithmetic of elliptic curves , GTM 106 , Springer-Verlag. · Zbl 0585.14026 |

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