Fourier expansions at cusps. (English) Zbl 1468.11099
In the paper under review, the authors concern the fields generated by the Fourier coefficients of modular forms at arbitrary cusps.
The main tools that are used in the paper are the action of GL\(_2^+ (\mathbb{Q})\) via the slash-operator and the action of Aut\((\mathbb{C})\) on the Fourier coefficients of a modular form and it is followed Shimura’s ideas and they give a new proof of Shimura’s formula for the action of Aut\((\mathbb{C})\) on \(f \mid g\) by using a theorem of Khuri-Makdisi on products of Eisenstein series.
They prove that these fields are contained in certain cyclotomic extensions of the field generated by the Fourier coefficients at infinity. They also show that this bound is tight in the case of newforms with trivial Nebentypus.
The paper concludes with three open research questions.
The main tools that are used in the paper are the action of GL\(_2^+ (\mathbb{Q})\) via the slash-operator and the action of Aut\((\mathbb{C})\) on the Fourier coefficients of a modular form and it is followed Shimura’s ideas and they give a new proof of Shimura’s formula for the action of Aut\((\mathbb{C})\) on \(f \mid g\) by using a theorem of Khuri-Makdisi on products of Eisenstein series.
They prove that these fields are contained in certain cyclotomic extensions of the field generated by the Fourier coefficients at infinity. They also show that this bound is tight in the case of newforms with trivial Nebentypus.
The paper concludes with three open research questions.
Reviewer: Ilker Inam (Bilecik)
MSC:
| 11F11 | Holomorphic modular forms of integral weight |
| 11F30 | Fourier coefficients of automorphic forms |
| 11R18 | Cyclotomic extensions |
Keywords:
Fourier coefficients; modular forms; cusps; newforms; Atkin-Lehner operators; cyclotomic extensionsReferences:
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