On local-global divisibility by $p^n$ in elliptic curves

TL;DR

This paper proves that for certain elliptic curves over number fields, the local-global divisibility principle by powers of a prime holds if no specific torsion points exist, with results depending on the field and prime size.

Contribution

It establishes conditions under which the local-global divisibility principle holds for elliptic curves over number fields, extending previous results with explicit bounds.

Findings

No counterexamples for large primes p depending on the degree of k

Local-global principle holds if no k-rational torsion points of order p exist

Explicit bounds for elliptic curves over fields of degree up to 5

Abstract

Let $p$ be a prime number and let $ k $ be a number field, which does not contain the field $\mathbb{Q} (\zeta_p + \bar{\zeta_p})$. Let $\mathcal{E}$ be an elliptic curve defined over $k$. We prove that if there are no $k$-rational torsion points of exact order $p$ on $\E$, then the local-global principle holds for divisibility by $p^n$, with $n$ a natural number. As a consequence of the deep theorem of Merel, for $p$ larger than a constant depending only on the degree of $k$, there are no counterexamples to the local-global divisibility principle. Nice and deep works give explicit small constants for elliptic curves defined over a number field of degree at most 5 over $\mathbb{Q}.