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We prove that the existence of a counterexample to the local-global divisibility by $p^2$ in $\\mathcal{E}$, assures the existence of a $k$-rational point of exact order $p$ in $\\mathcal{E}$. Using the Merel Theorem, we then shrunk the known set of primes for which there could be a counterexample to the local-global divisibility by $p^2$."},"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":"1103.4963","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-03-25T13:18:28Z","cross_cats_sorted":[],"title_canon_sha256":"3588e6b441a3f3441e37a46beae7851515921d840bdb35b8196dec2061e178ba","abstract_canon_sha256":"cd36f1fe8777c12896a0c07a68ff6346ac96daa4b1369578a906d1ac7d4f5476"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:25:48.299972Z","signature_b64":"R88/QsV4/7NaD2R8PlFOt75KQ9rThwqlEhFSEYS18ruN9xljrko8zKPG8S6S8P/BtIj29Pr8SGQKHM/Lzxy3Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e6d8bd5b62ed468efd64f46f1cd4ab53e942ac9fafba8cd093f87ea3013e0005","last_reissued_at":"2026-05-18T04:25:48.299470Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:25:48.299470Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On local-global divisibility by $p^2$ in elliptic curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Evelina Viada, Gabriele Ranieri, Laura Paladino","submitted_at":"2011-03-25T13:18:28Z","abstract_excerpt":"Let $ p $ be a prime lager than 3. 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