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On generation of the coefficient field of a primitive Hilbert modular form by a single Fourier coefficient

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  08 September 2022

Narasimha Kumar Open the ORCID record for Narasimha Kumar [Opens in a new window]  and
Satyabrat Sahoo Open the ORCID record for Satyabrat Sahoo [Opens in a new window]
Narasimha Kumar*
Affiliation:
Department of Mathematics, Indian Institute of Technology Hyderabad, Sangareddy 502285, India e-mail: ma18resch11004@iith.ac.in
Satyabrat Sahoo
Affiliation:
Department of Mathematics, Indian Institute of Technology Hyderabad, Sangareddy 502285, India e-mail: ma18resch11004@iith.ac.in
*
e-mail: narasimha@math.iith.ac.in
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Abstract

Let f be a primitive Hilbert modular form over F of weight k with coefficient field $E_f$, generated by the Fourier coefficients $C(\mathfrak {p}, f)$ for $\mathfrak {p} \in \mathrm {Spec}(\mathcal {O}_F)$. Under certain assumptions on the image of the residual Galois representations attached to f, we calculate the Dirichlet density of $\{\mathfrak {p} \in \mathrm {Spec}(\mathcal {O}_F)| E_f = \mathbb {Q}(C(\mathfrak {p}, f))\}$. For $k=2$, we show that those assumptions are satisfied when $[E_f:\mathbb {Q}] = [F:\mathbb {Q}]$ is an odd prime. We also study analogous results for $F_f$, the fixed field of $E_f$ by the set of all inner twists of f. Then, we provide some examples of f to support our results. Finally, we compute the density of $\{\mathfrak {p} \in \mathrm {Spec}(\mathcal {O}_F)| C(\mathfrak {p}, f) \in K\}$ for fields K with $F_f \subseteq K \subseteq E_f$.

Keywords

Hilbert modular formsFourier coefficientsfinite generationdensityinner twists

MSC classification

Primary: 11F30: Fourier coefficients of automorphic forms
Secondary: 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F80: Galois representations

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 2 , June 2023 , pp. 587 - 598
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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