The local-global principle for divisibility in CM elliptic curves

Abstract

We consider the local-global principle for divisibility in the Mordell-Weil group of a CM elliptic curve defined over a number field. For each prime p we give lower bounds on the degree d of a number field over which there exists a CM elliptic curve which gives a counterexample to the local-global principle for divisibility by a power of p. As a corollary we deduce that there are at most finitely many elliptic curves (with or without CM) which are counterexamples with p>2d+1. We also deduce that the local-global principle for divisibility by powers of 7 holds for all elliptic curves over quadratic fields.