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

Kumar, Narasimha  ; Sahoo, Satyabrat 

Canadian mathematical bulletin, Tome 66 (2023) no. 2, pp. 587-598

Résumé

Let 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 $F_f \subseteq K \subseteq E_f$.

Détail

DOI : 10.4153/S0008439522000558

Mots-clés : Hilbert modular forms, Fourier coefficients, finite generation, density, inner twists