Modularity of CM elliptic curves over division fields (Q958661)

Let \(E\) be an elliptic curve defined over an algebraic number field \(F\) such that \(\text{End}_{\overline Q}(E)\) is isomorphic to an order \(R\) of an imaginary quadratic field. One knows that there exists a normalized new form \(f\) of weight two on \(\Gamma_1(N)\) for some \(N\) such that \(E\) admits a non-zero homomorphism \(\varphi:E\to J_f\) defined over \(\overline Q\), where \(J_f\) is the \(Q\)-simple factor of the Jacobian variety of \(\Gamma_1(N)\) corresponding to \(f\). If the homomorphism \(\varphi\) is defined over a field \(K\), we say \(E\) is modular over \(K\). In the author's previous paper [J. Number Theory 108, No. 2, 268--286 (2004; Zbl 1079.11030)], he gave a criterion for \(E\) to be modular over \(F\). In the present paper the author proves among other things the following: Let \(p\) be a prime number and \({\mathfrak p}\) a prime ideal of \(R\) lying above \(p\), and assume that \(p\) is prime to the order of the unit group of \(R\). Then \(E\) is modular over \(F(E[{\mathfrak p}]) \), the field obtained by adjoining to \(F\) the coordinates of \({\mathfrak p}\)-division points of \(E\).