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

Abstract

AbstractLet f be a primitive Hilbert modular form over F of weight k with coefficient field $E_f$ , generated by the Fourier coefficients $C(\mathfrak {p}, f)$ for $\mathfrak {p} \in \mathrm {Spec}(\mathcal {O}_F)$ . Under certain assumptions on the image of the residual Galois representations attached to f, we calculate the Dirichlet density of $\{\mathfrak {p} \in \mathrm {Spec}(\mathcal {O}_F)| E_f = \mathbb {Q}(C(\mathfrak {p}, f))\}$ . For $k=2$ , we show that those assumptions are satisfied when $[E_f:\mathbb {Q}] = [F:\mathbb {Q}]$ is an odd prime. We also study analogous results for $F_f$ , the fixed field of $E_f$ by the set of all inner twists of f. Then, we provide some examples of f to support our results. Finally, we compute the density of $\{\mathfrak {p} \in \mathrm {Spec}(\mathcal {O}_F)| C(\mathfrak {p}, f) \in K\}$ for fields K with $F_f \subseteq K \subseteq E_f$ .