On the Finiteness of the Number of Elliptic Fields with Given Degrees of \(S\)-Units and Periodic Expansion of \(\sqrt f \)

Abstract

For a field k of characteristic 0, up to a natural equivalence relation, it is proved that the number of nontrivial elliptic fields \(k(x)(\sqrt f )\) with a periodic continued fraction expansion of \(\sqrt f \in k((x))\) for which the corresponding elliptic curve contains a k-point of even order at most 18 or a k-point of odd order at most 11 is finite. In the case when k is a quadratic extension of \(\mathbb{Q}\), all such fields are found.