On the number of elliptic curves with CM over large algebraic fields (Q818287)

Let \(\mathbb Q\) be the rational number field and let Gal\(({\mathbb Q})\) be its absolute Galois group. For any integer \(e\geq{1}\), let Gal\(({\mathbb Q})^{e}\) be the Cartesian power. Let \(\widetilde{\mathbb Q}\) be the algebraic closure of \(\mathbb Q\). For each \({\sigma}=(\sigma_1, \sigma_2, ..., \sigma_{n})\), let \(\widetilde{Q({\sigma})}\) be the fixed field in \(\widetilde{\mathbb Q}\) of \(\sigma_1, \sigma_2, ..., \sigma_{n}\). For each \({\sigma}\in \)Gal\(({\mathbb Q})^{e}\), we want to know the number of the elliptic curves \(E\) (up to \(C\)-isomorphism) with CM that are defined over \(\widetilde{\mathbb Q({\sigma})}\). We are also interested in knowing the number of elliptic curves \(E\) (up to \(C\)-isomorphism) with CM that are defined over \(\widetilde{\mathbb Q({\sigma})}\), and such that all \(C\)-endomorphisms of \(E\) are defined over \(\widetilde{\mathbb Q({\sigma})}\). The authors cleverly use several fundamental results from their previous work and other authors, to answer these questions. In fact, they show there is a positive integer \(e_0\) such that there are infinitely many elliptic curves \(E\) (up to \(C\)-isomorphism) with CM over \(\widetilde{\mathbb Q({\sigma})}\) if and only if \(e\leq{e_0}\). The authors specifically prove \(e_0=3\) for the first task and \(e_0=2\) for the second one.