Hostname: page-component-6bf8c574d5-7jkgd Total loading time: 0 Render date: 2025-02-28T09:20:26.211Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barnet-Lamb, T., Gee, T., and Geraghty, D., The Sato–Tate conjecture for Hilbert modular forms. J. Amer. Math. Soc. 24(2011), no. 2, 411–469.CrossRefGoogle Scholar
Carayol, H., Sur les représentations $l$ -adiques associées aux formes modulaires de Hilbert (in French) [On $l$ -adic representations associated with Hilbert modular forms]. Ann. Sci. École Norm. Supér. (4) 19(1986), no. 3, 409–468.CrossRefGoogle Scholar
Dalal, T. and Kumar, N., On non-vanishing and sign changes of the Fourier coefficients of Hilbert cusp forms. Topics in number theory. Ramanujan Math. Soc. Lect. Notes Ser. 26(2020), 175–188.Google Scholar
Dieulefait, L. and Dimitrov, M., Explicit determination of images of Galois representations attached to Hilbert modular forms. J. Number Theory 117(2006), 397–405.CrossRefGoogle Scholar
Dimitrov, M., Galois representations modulo p and cohomology of Hilbert modular varieties. Ann. Sci. École Norm. Supér. (4) 38(2005), no. 4, 505–551.10.1016/j.ansens.2005.03.005CrossRefGoogle Scholar
Koo, K. T.-L., Stein, W., and Wiese, G., On the generation of the coefficient field of a newform by a single Hecke eigenvalue. J. Theor. Nombres Bordeaux 20(2008), no. 2, 373–384.CrossRefGoogle Scholar
Marcus, D. A., Number fields, Universitext, Springer, New York–Heidelberg, 1977.CrossRefGoogle Scholar
Ribet, K. A., On l-adic representations attached to modular forms. II. Glasg. Math. J. 27(1985), 185–194.CrossRefGoogle Scholar
Serre, J.-P., Quelques applications du théorème de densité de Chebotarev (in French) [ Some applications of the Chebotarev density theorem ]. Publ. Math. Inst. Hautes Etudes Sci. 54(1981), 323–401.CrossRefGoogle Scholar
Shemanske, T. R. and Walling, L. H., Twists of Hilbert modular forms. Trans. Amer. Math. Soc. 338(1993), no. 1, 375–403.CrossRefGoogle Scholar
Shimura, G., The special values of the zeta functions associated with Hilbert modular forms. Duke Math. J. 45(1978), no. 3, 637–679.CrossRefGoogle Scholar
Taylor, R., On Galois representations associated to Hilbert modular forms. Invent. Math. 98(1989), no. 2, 265–280.CrossRefGoogle Scholar
The LMFDB Collaboration, The L-functions and Modular Forms Database, 2021. http://www.lmfdb.org [accessed September 7, 2022].Google Scholar