On generation of the coefficient field of a primitive Hilbert modular form by a single Fourier coefficient

TL;DR

This paper investigates the distribution of primes for which Fourier coefficients of primitive Hilbert modular forms generate their coefficient fields, extending previous results and providing explicit density calculations under certain conditions.

Contribution

It introduces new density formulas for primes generating coefficient fields of primitive Hilbert modular forms, generalizing prior work to a broader class of forms.

Findings

Density of primes generating the coefficient field is explicitly calculated.

Conditions under which Fourier coefficients generate the field are established.

Examples support the theoretical density results.

Abstract

For a primitive Hilbert modular form $f$ over $F$ of weight $k$, under certain assumptions on image of $\bar{\rho}_{f,\lambda}$, we calculate the Dirichlet density of primes $\mathfrak{p}$ for which the $\mathfrak{p}$-th Fourier coefficient $C(\mathfrak{p}, f)$ generates the coefficient field $E_f$. If $k=2$, then we show that the assumption on the image of $\bar{\rho}_{f,\lambda}$ is satisfied when the degrees of $E_f, F$ are equal and odd prime. We also compute the density of primes $\mathfrak{p}$ for which $C^*(\mathfrak{p}, f)$ generates $F_f$. Then, we provide some examples of $f$ to support our results. Finally, we calculate the density of primes $\mathfrak{p}$ for which $C(\mathfrak{p}, f) \in K$ for any field $K$ with $F_f \subseteq K \subseteq E_f$. This density is completely determined by the inner twists of $f$ associated with $K$. This work can be thought of as a generalization of~\cite{KSW08} to primitive Hilbert modular forms.