On the local-global divisibility of torsion points on elliptic curves and ${\rm GL}_2$-type varieties

TL;DR

This paper proves that for odd primes, local-global divisibility by any power of p holds for torsion points on elliptic curves over number fields, and extends some results to GL2-type varieties and abelian surfaces.

Contribution

It establishes the local-global divisibility for torsion points on elliptic curves for odd primes and generalizes the result to GL2-type varieties and certain abelian surfaces.

Findings

Local-global divisibility holds for torsion points on elliptic curves when p is odd.

Counterexamples show the prime hypothesis is necessary.

Results extend to GL2-type varieties and abelian surfaces with quaternionic multiplication.

Abstract

Let $p$ be a prime number and let $k$ be a number field. Let $E$ be an elliptic curve defined over $k$. We prove that if $p$ is odd, then the local-global divisibility by any power of $p$ holds for the torsion points of $E$. We also show with an example that the hypothesis over $p$ is necessary. We get a weak generalization of the result on elliptic curves to the larger family of ${\rm GL}_2$-type varieties over $k$. In the special case of the abelian surfaces $A/k$ with quaternionic multiplication over $k$ we obtain that for all prime $p$, except a finite number depending on $A$, the local-global divisibility by any power of $p$ holds for the torsion points of $A$