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

Kumar, Narasimha and Sahoo, Satyabrat (2022) On generation of the coefficient field of a primitive Hilbert modular form by a single Fourier coefficient. Canadian Mathematical Bulletin. pp. 1-12. ISSN 0008-4395

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Abstract

Let f be a primitive Hilbert modular form over F of weight k with coefficient field E-f, generated by the Fourier coefficients C(p, f) for p is an element of Spec(O-F). Under certain assumptions on the image of the residual Galois representations attached to f, we calculate the Dirichlet density of {p is an element of Spec(O-F)vertical bar E-f = Q(C(p, f))}. For k = 2, we show that those assumptions are satisfied when [E-f : Q] = [ F : 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 {p is an element of Spec(O-F)vertical bar C(p, f) is an element of K} for fields K with F-f subset of K subset of E-f.

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