Density zero results for elliptic curves without complex multiplication (Q1127561)

Let \(n\geq 2\) be an integer, \(K\) a number field, \(E\) an elliptic curve defined over \(K\) and let \(C\) be a cyclic subgroup of \(E\) of order \(n\), which is invariant under all \(K\)-automorphisms of \(\overline{\mathbb{Q}}\), the algebraic closure of \(\mathbb{Q}\). Let \({\mathcal E}_n^{\text{CM}}(K)=\{(E,C)\mid E\) has complex multiplication and \(C\) as above\}, \({\mathcal E}_n^{\text{no CM}}(K)= \{(E,C)\mid E\) has no complex multiplication and \(C\) as above\}, and let \({\mathcal E}_n(K)= {\mathcal E}_n^{\text{CM}}(K)\cup {\mathcal E}_n^{\text{no CM}}(K)\). For \(K=\mathbb{Q}\), the following complete result is true: \({\mathcal E}_n (\mathbb{Q})\neq \emptyset\) iff \(n\leq 19\) or \(n=21\), 25, 27, 37, 43, 67, 163. It is also known that if \(k\) is an imaginary quadratic field of class number 1, and \(p\equiv 1(4)\) is prime, then \({\mathcal E}_n^{\text{CM}}(K)\neq \emptyset\). The author conjectures that for every number field \(K\), there is an integer \(n_0(K)\), such that for \(n>n_0(K)\), \({\mathcal E}_n^{\text{no CM}}(K)= \emptyset\). The main result of the paper is: The set \({\mathcal N}(K)= \{n\geq 2\mid {\mathcal E}_n^{\text{no CM}} (K)\neq \emptyset\}\) has uniform density 0 for every number field \(K\). The author recalls the notion of uniform density and proves further results and corollaries.