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Decomposition theorems for Hilbert modular forms. (English) Zbl 1278.11054
Summary: 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.

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
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