Functiones et Approximatio Commentarii Mathematici

Decomposition theorems for Hilbert modular forms

Benjamin Linowitz

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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.

Article information

Source
Funct. Approx. Comment. Math., Volume 47, Number 2 (2012), 157-172.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.facm/1356012913

Digital Object Identifier
doi:10.7169/facm/2012.47.2.3

Mathematical Reviews number (MathSciNet)
MR3051446

Zentralblatt MATH identifier
1278.11054

Subjects
Primary: 11F41: Automorphic forms on GL(2); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20]

Keywords
Hilbert modular form newform character twist

Citation

Linowitz, Benjamin. Decomposition theorems for Hilbert modular forms. Funct. Approx. Comment. Math. 47 (2012), no. 2, 157--172. doi:10.7169/facm/2012.47.2.3. https://projecteuclid.org/euclid.facm/1356012913


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