EXCEPTIONAL ELLIPTIC CURVES OVER QUARTIC FIELDS

We study the number of elliptic curves, up to isomorphism, over a fixed quartic field K having a prescribed torsion group T as a subgroup. Let T = Z/mZ circle plus Z/nZ, where m vertical bar n, be a torsion group such that the modular curve X-1(m, n) is an elliptic curve. Let K be a number field such that there is a positive and finite number of elliptic curves E over K having T as a subgroup. We call such pairs (T, K) exceptional. It is known that there are only finitely many exceptional pairs when K varies through all quadratic or cubic fields. We prove that when K varies through all quartic fields, there exist infinitely many exceptional pairs when T = Z/14Z or Z/15Z and finitely many otherwise.