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December 2012 Decomposition theorems for Hilbert modular forms
Benjamin Linowitz
Funct. Approx. Comment. Math. 47(2): 157-172 (December 2012). DOI: 10.7169/facm/2012.47.2.3

Abstract

Let $\mathscr{S}_k^+(\mathcal{N},\Phi)$ 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 $\mathscr{S}_k^+(\mathcal{N},\Phi)$ 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 $\mathscr{S}_k^+(\mathcal{N},\Phi)$ 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 $\mathscr{S}_k^+(\mathcal{N},\Phi)$ 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.

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Benjamin Linowitz. "Decomposition theorems for Hilbert modular forms." Funct. Approx. Comment. Math. 47 (2) 157 - 172, December 2012. https://doi.org/10.7169/facm/2012.47.2.3

Information

Published: December 2012
First available in Project Euclid: 20 December 2012

zbMATH: 1278.11054
MathSciNet: MR3051446
Digital Object Identifier: 10.7169/facm/2012.47.2.3

Subjects:
Primary: 11F41

Rights: Copyright © 2012 Adam Mickiewicz University

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Vol.47 • No. 2 • December 2012
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