On the Hasse principle for divisibility in elliptic curves
Jessica Alessandrì, Laura Paladino
TL;DR
This paper addresses the local-global divisibility problem for elliptic curves: determining when local divisibility by $p^n$ implies global divisibility by $p^n$ for $n\ge 2$. It develops a cohomological framework centered on the first local cohomology group $H^1_{\rm loc}(G_n,{\mathcal E}[p^n])$, showing its vanishing is equivalent to an affirmative local-global principle and to $\Sha(k,{\mathcal E}[p^n])=0$. The authors derive explicit sufficient conditions on the generators of the division-field Galois group $G_n={\rm Gal}(k({\mathcal E}[p^n])/k)$, decomposing $G_n$ into diagonal, upper, and lower triangular components, which guarantee $H^1_{\rm loc}(G_n,{\mathcal E}[p^n])=0$ for all $n\ge 2$, hence the local-global divisibility holds; these conditions also yield $\Sha(k,{\mathcal E}[p^n])=0$ and a positive answer to the second local-global problem. The paper then shows these hypotheses are sharp by constructing counterexamples for $p\ge 5$ and all $n\ge 2$ where $H^1_{\rm loc}(G_n,{\mathcal E}[p^n])\neq 0$, demonstrating failures of the local-global divisibility, and extending these counterexamples to higher powers over finite extensions. Overall, the work generalizes Ranieri’s $n=2$ results to all powers $p^n$ and clarifies the group-theoretic structure that governs the local-global behavior in elliptic curves.
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 $n=2$.