The local-global principle for divisibility in CM elliptic curves

TL;DR

This paper investigates when the local-global principle for divisibility in the Mordell-Weil group of CM elliptic curves over number fields fails, providing bounds and specific cases where it holds or fails.

Contribution

It establishes sharp lower bounds on degrees of number fields for counterexamples and proves the principle holds for powers of 7 over quadratic fields.

Findings

Finiteness of counterexamples with large primes

Counterexamples exist only over fields of bounded degree

Local-global principle holds for powers of 7 over quadratic fields

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 sharp 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 over quadratic fields.