Fourier coefficients of Hilbert modular forms at cusps

TL;DR

This paper investigates the algebraic fields generated by Fourier coefficients of Hilbert modular forms at various cusps, extending understanding of their arithmetic properties and field extensions.

Contribution

It provides a description of the fields generated by Fourier coefficients at arbitrary cusps, including explicit cyclotomic extensions containing these coefficients.

Findings

Fourier coefficients at cusps lie in specific cyclotomic extensions.

The field generated by coefficients at infinity is contained in a cyclotomic extension.

The results generalize previous knowledge from the cusp at infinity to arbitrary cusps.

Abstract

The aim of this article is to study the fields generated by the Fourier coefficients of Hilbert newforms at arbitrary cusps. Precisely, given a cuspidal Hilbert newform $f$ and a matrix $\sigma$ in (a suitable conjugate of) the Hilbert modular group, we give a cyclotomic extension of the field generated by the Fourier coefficients at infinity which contains all the Fourier coefficients of $f||_k\sigma$.