Fourier expansions at cusps

François Brunault (1) , Michael Neururer

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

In this article we study the number 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, and show that this bound is tight in the case of newforms with trivial Nebentypus. The main tool is an extension of a result of Shimura on the interplay between the actions of SL_2(Z) and Aut(C) on spaces of modular forms. We give two new proofs of this result: one based on products of Eisenstein series, and the other using the theory of algebraic modular forms.

Domaines: Mathématiques [math]; Théorie des nombres [math.NT]

François Brunault, Michael Neururer. Fourier expansions at cusps. Ramanujan Journal, 2020, 53, pp.423-437. ⟨10.1007/s11139-019-00178-5⟩. ⟨ensl-01953138v2⟩