An analogue of the Siegel-Walfisz theorem for the cyclicity of CM elliptic curves mod \(p\) (Q981850)

Let \(E\) be a CM elliptic curve defined over \({\mathbb Q}\) of conductor \(N\) and with complex multiplication by the full ring of integers of an imaginary quadratic field. Let \(C(x,N,E)\) be the number of primes \(p\leq x\) sucht that \(p\nmid N\) and the reduction \(\bmod\, p\) of \(E\) is cyclic. Let \(A,B>0\) and \(N\leq (\log x)^A\). The auhors prove that \[ C(x,N,E)={\mathfrak c}_E\text{li}(x)+O_{A,B}\left(\frac{x}{(\log x)^B}\right), \] uniformly in \(N\), where \[ {\mathfrak c}_E=\sum_{k=1}^\infty\frac{\mu(k)}{[{\mathbb Q}(E[k]):{\mathbb Q}]}, \] and the implied constant depends only on \(A\) and \(B\). Here \(\mu\) is the Möbius function. This is an analogue of the classical Siegel-Walfisz theorem on primes in arithmetic progressions.