Ranks of Elliptic Curves with Prescribed Torsion over Number Fields

Abstract

We study the structure of the Mordell--Weil group of elliptic curves over number fields of degree 2, 3, and 4. We show that if $T$ is a group, then either the class of all elliptic curves over quadratic fields with torsion subgroup $T$ is empty, or it contains curves of rank~0 as well as curves of positive rank. We prove a similar but slightly weaker result for cubic and quartic fields. On the other hand, we find a group $T$ and a quartic field $K$ such that among the elliptic curves over $K$ with torsion subgroup $T$, there are curves of positive rank, but none of rank~0. We find examples of elliptic curves with positive rank and given torsion in many previously unknown cases. We also prove that all elliptic curves over quadratic fields with a point of order 13 or 18 and all elliptic curves over quartic fields with a point of order 22 are isogenous to one of their Galois conjugates and, by a phenomenon that we call \emph{false complex multiplication}, have even rank. Finally, we discuss connections with elliptic curves over finite fields and applications to integer factorization.