On the torsion subgroups of elliptic curves over toally real fields (Q1071080)

Let K be a number field, and let \(K_ G\) be the smallest extension of K that is galois over \({\mathbb{Q}}\). Let \(G=Gal(K_ G/{\mathbb{Q}})\), and let \(d=[K_ G:{\mathbb{Q}}]\) denote the order of G. Finally, let E be an elliptic curve defined over K, and let \(E_{/K_ G}\) denote the base change of E to \(K_ G\). We say that \(E_{/K}\) is a G-curve if whenever one prime of \(K_ G\) of residue characteristic q divides the conductor of \(E_{/K_ G}\) then all primes of residue characteristic q divide the conductor of \(E_{/K_ G}.\) The author proves the following theorems. Theorem 3.1. Let K be a totally real field. There is a finite set \(S_ K\) of prime numbers such that if \(E_{/K}\) is a G-curve with a K- rational point of order p then \(p\in S_ K.\) Theorem 4.1. Let \(p^ n\) be the largest power of p occurring among the orders of the cyclic subgroups of \(K_ 2{\mathcal O}\). If E is a G-curve with a K-rational point of order \(p^ m\) then \(m\leq n\).