Cyclicity and generation of points mod \(p\) on elliptic curves (Q807692)

Let \(E\) be an elliptic curve defined over \({\mathbb Q}\) and, for each prime \(p\) for which \(E\) has good reduction, let \(E({\mathbb F}_ p)\) be the group of rational points on the reduction of \(E\) modulo \(p\). First Serre raised the question how often this group is cyclic and proved, assuming the generalized Riemann hypothesis (GRH), that the set of primes with this property has a positive density. Using results of sieve theory the authors succeed in proving a similar result without the assumption of the GRH. This result is a generalization of a previous result of the second author [J. Number Theory 16, 147--168 (1983; Zbl 0526.12010)]. Now let \(\Gamma\) be a free subgroup of \(E({\mathbb Q})\) of rank \(r\) and define \(S(\Gamma)=\{p\text{ prime}\mid E({\mathbb F}_ p)=\Gamma_ p\},\) where \(\Gamma_ p\) is the reduction of \(\Gamma\) mod \(p\) and, if \(E\) has CM by an imaginary quadratic field \(K\), \(S'(\Gamma)=\{p\text{ prime}\mid E({\mathbb F}_ p)=\Gamma_ p \text{ and } p \text{ is inert in } K\}\). Under the assumption of the GRH and if \(r\) is sufficiently large then \(S(\Gamma)\), \(S'(\Gamma)\) have certain densities. This result is also an improvement of the authors' previous work in Compos. Math. 58, 13--44 (1986; Zbl 0598.14018). The whole problem is related to Artin's primitive root conjecture for elliptic curves [cf. \textit{S. Lang} and \textit{H. Trotter}, Bull. Am. Math.Soc. 83, 289--292 (1977; Zbl 0345.12008)].