{"paper":{"title":"On the local-global divisibility over ${\\rm GL}_2$-type varieties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Florence Gillibert, Gabriele Ranieri","submitted_at":"2017-03-18T02:37:50Z","abstract_excerpt":"Let $k$ be a number field and let ${\\mathcal{A}}$ be a ${\\rm GL}_2$-type variety defined over $k$ of dimension $d$. We show that for every prime number $p$ satisfying certain conditions (see Theorem 2), if the local-global divisibility principle by a power of $p$ does not hold for ${\\mathcal{A}}$ over $k$, then there exists a cyclic extension $\\widetilde{k}$ of $k$ of degree bounded by a constant depending on $d$ such that ${\\mathcal{A}}$ is $\\widetilde{k}$-isogenous to a ${\\rm GL}_2$-type variety defined over $\\widetilde{k}$ that admits a $\\widetilde{k}$-rational point of order $p$. Moreover,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.06235","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"}