On the Hasse principle for divisibility in elliptic curves
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| Format: | Preprint |
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2025
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| author | Alessandrì, Jessica Paladino, Laura |
| author_facet | Alessandrì, Jessica Paladino, Laura |
| contents | 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 $n=2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_02078 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Hasse principle for divisibility in elliptic curves Alessandrì, Jessica Paladino, Laura Number Theory 11R34, 1G05, 14K02, 14G05 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 $n=2$. |
| title | On the Hasse principle for divisibility in elliptic curves |
| topic | Number Theory 11R34, 1G05, 14K02, 14G05 |
| url | https://arxiv.org/abs/2511.02078 |