On the Hasse principle for divisibility in elliptic curves

TL;DR

This paper investigates the conditions under which the local-global divisibility principle holds for elliptic curves over number fields, especially for powers of primes, providing new sufficient and necessary conditions and counterexamples.

Contribution

It establishes new criteria on Galois groups that determine the validity of the Hasse principle for divisibility by prime powers in elliptic curves, extending previous results to all powers $p^n$.

Findings

Conditions on Galois groups imply divisibility holds for all $n \\geq 2$

Counterexamples are constructed when conditions fail

Results generalize previous work for $n=2$

Abstract

Let $p$ be a prime number and $n$ a positive integer. Let $E$ be an elliptic curve defined over a number field $k$. It is known that the local-global divisibility by $p$ holds in $E/k$, but for powers of $p^n$ counterexamples may appear. The validity or the failing of the Hasse principle depends on the elliptic curve $E$ and the field $k$ and, consequently, on the group $\mathrm{Gal}(k(E[p^n])/k)$. For which kind of these groups does the principle hold? For which of them can we find a counterexample? The answer to these questions was known for $n=1,2$, but for $n\geq 3$ they were still open. We show some conditions on the generators of $\mathrm{Gal}(k(E[p^n])/k)$ implying an affirmative answer to the local-global divisibility by $p^n$ in $E$ over $k$, for every $n\geq 2$. We also prove that these conditions are necessary by producing counterexamples in the case when they do not hold. These last results generalize to every power $p^n$, a result obtained by Ranieri for $n=2$.