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We also show with an example that the hypothesis over $p$ is necessary.\n  We get a weak generalization of the result on elliptic curves to the larger family of ${\\rm GL}_2$-type varieties over $k$. In the special case of the abelian surfaces $A/k$ with quaternionic multiplication over $k$ we obtain that for all prime $p$, except a finite number depending on $A$, the local-"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1609.00410","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-09-01T21:49:38Z","cross_cats_sorted":[],"title_canon_sha256":"8b55c18c04537da4c984747eb30494b3301d855bc944dfdaf8de4ac9c0810daf","abstract_canon_sha256":"19ac7a17e159dd02085521c96a23df74e6fb8f5eee3e8766fb70ef31512543f1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:06:23.935011Z","signature_b64":"A2YCjqfd6jOylwWKZgqfSNi8ntR7O1DXc99twUJf3IrmvFUplcohw42/TR5RN3i9hrzJY2T8CfmywurXbrRiAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7617e0787ca6c13282d6eb2958fb4ae06cf94f00986acd960fde11f05903834d","last_reissued_at":"2026-05-18T01:06:23.934557Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:06:23.934557Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the local-global divisibility of torsion points on elliptic curves and ${\\rm GL}_2$-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":"2016-09-01T21:49:38Z","abstract_excerpt":"Let $p$ be a prime number and let $k$ be a number field. 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