Fourier expansions at cusps

TL;DR

This paper investigates the algebraic fields generated by Fourier coefficients of modular forms at various cusps, establishing their containment within cyclotomic extensions and demonstrating the bounds' tightness for certain newforms.

Contribution

It proves that fields generated by Fourier coefficients at arbitrary cusps are contained in cyclotomic extensions, refining understanding of their algebraic structure and bounds.

Findings

Fourier coefficient fields at cusps are within cyclotomic extensions.

The bounds on these fields are tight for newforms with trivial Nebentypus.

Shimura's results are used to analyze the Galois actions on modular forms.

Abstract

In this article we study the fields generated by the Fourier coefficients of modular forms at arbitrary cusps. We prove that these fields are contained in certain cyclotomic extensions of the field generated by the Fourier coefficients at infinity. We also show that this bound is tight in the case of newforms with trivial Nebentypus. The main tool is a result of Shimura on the interplay between the actions of $\mathrm{GL}_2^+(\mathbb{Q})$ and $\mathrm{Aut}(\mathbb{C})$ on modular forms.