{"paper":{"title":"Counterexamples to the local-global divisibility over elliptic curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Gabriele Ranieri","submitted_at":"2017-05-04T15:30:44Z","abstract_excerpt":"Let $p \\geq 5$ be a prime number. We find all the possible subgroups $G$ of ${\\rm GL}_2 ( \\mathbb{Z} / p \\mathbb{Z} )$ such that there exists a number field $k$ and an elliptic curve ${\\mathcal{E}}$ defined over $k$ such that the ${\\rm Gal} ( k ( {\\mathcal{E}}[p] ) / k )$-module ${\\mathcal{E}}[p]$ is isomorphic to the $G$-module $( \\mathbb{Z} / p \\mathbb{Z} )^2$ and there exists $n \\in \\mathbb{N}$ such that the local-global divisibility by $p^n$ does not hold over ${\\mathcal{E}} ( k )$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.01880","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}