Density zero results for elliptic curves without complex multiplication

Abstract.

It is conjectured that if K is any number field, there exists a positive integer \( n_0(K) \) such that if \( n>n_0(K) \) the following set is empty: \( {\cal E}_{n}^{noCM}(K)=\{(E,C)| E \) is an elliptic curve defined over K, without complex multiplication, C is a cyclic subgroup of order q, invariant by K-automorphisms of \( \overline {\Bbb Q} \). Let \( {\cal N}(K)=\{n>2|{\cal E}_{n}^{noCM}(K)\ne \emptyset \} \). We prove that for every K the set \( {\cal N} (K)\) has uniform density 0.