On the field of definition for modularity of CM elliptic curves (Q1773333)

Let \(E\) be CM elliptic curve defined over a number field \(F\) with complex multiplication by an imaginary quadratic field \(K\). It is known that there exists a new form \(f\) such that \(\Hom_{\overline {\mathbb Q}}(E,J_f)\neq\{0\}\), where \(J_f\) denotes the abelian variety over \(\mathbb Q\) associated to \(f\) by the Eichler-Shimura theory. In this paper, the author considers the problem to determine the field over which the homomorphisms are defined, and obtains a criterion for \(\Hom_F(E,J_f)\) to be nontrivial. Let \(F'= FK\) and \(\beta_{E/F'}: F^{\prime\times}_{\mathbb A}\to \mathbb C^\times\) denote the Grössen-character of \(E/F'\). He shows that \(\Hom_F(E,J_f)\neq\{0\}\) if and only if there exists a Grössen-character \(\gamma: K^\times_{\mathbb A}\to\mathbb C\) such that \(\gamma\circ N_{F'/K}= \beta_{E/F'}\) and the latter is equivalent to the condition that every torsion element of \(E\) is rational over \(FK_{ab}\). Furthermore, he determines the isogeny decomposition of \(J_f\) over \(F\) when these conditions are met.