{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2007:3ARB77OY4KCEWFEN3JRRLAUS4U","short_pith_number":"pith:3ARB77OY","schema_version":"1.0","canonical_sha256":"d8221ffdd8e2844b148dda63158292e532e43c33d723969bf85ba1f4f7532305","source":{"kind":"arxiv","id":"0712.2815","version":3},"attestation_state":"computed","paper":{"title":"Two variants of the support problem for products of abelian varieties and tori","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Antonella Perucca","submitted_at":"2007-12-17T20:58:32Z","abstract_excerpt":"Let G be the product of an abelian variety and a torus defined over a number field K. Let P and Q be K-rational points on G. Suppose that for all but finitely many primes p of K the order of (Q mod p) divides the order of (P mod p). Then there exist a K-endomorphism f of G and a non-zero integer c such that f(P)=cQ. Furthermore, we are able to prove the above result with weaker assumptions: instead of comparing the order of the points we only compare the radical of the order (radical support problem) or the l-adic valuation of the order for some fixed rational prime l (l-adic support problem)."},"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":"0712.2815","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2007-12-17T20:58:32Z","cross_cats_sorted":[],"title_canon_sha256":"f02d71509d26a86ce79f109792283b1758e7f1281edc3fa2522d61fe28909126","abstract_canon_sha256":"2e8126d4e7f9b8462045d3c3e81a9816fafef69960d54511a17f628844c4ef76"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:37:46.182385Z","signature_b64":"uEbxt8KoqI5k8edZ4oTwMe3AANfJzvhM0Sxbn+orSFkElwaAhLKNVGAlAH1/C5kuKfyKS1wM9CeGx1GzzgshCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d8221ffdd8e2844b148dda63158292e532e43c33d723969bf85ba1f4f7532305","last_reissued_at":"2026-07-04T15:37:46.181934Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:37:46.181934Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Two variants of the support problem for products of abelian varieties and tori","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Antonella Perucca","submitted_at":"2007-12-17T20:58:32Z","abstract_excerpt":"Let G be the product of an abelian variety and a torus defined over a number field K. Let P and Q be K-rational points on G. Suppose that for all but finitely many primes p of K the order of (Q mod p) divides the order of (P mod p). Then there exist a K-endomorphism f of G and a non-zero integer c such that f(P)=cQ. Furthermore, we are able to prove the above result with weaker assumptions: instead of comparing the order of the points we only compare the radical of the order (radical support problem) or the l-adic valuation of the order for some fixed rational prime l (l-adic support problem)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0712.2815","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/0712.2815/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0712.2815","created_at":"2026-07-04T15:37:46.182000+00:00"},{"alias_kind":"arxiv_version","alias_value":"0712.2815v3","created_at":"2026-07-04T15:37:46.182000+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0712.2815","created_at":"2026-07-04T15:37:46.182000+00:00"},{"alias_kind":"pith_short_12","alias_value":"3ARB77OY4KCE","created_at":"2026-07-04T15:37:46.182000+00:00"},{"alias_kind":"pith_short_16","alias_value":"3ARB77OY4KCEWFEN","created_at":"2026-07-04T15:37:46.182000+00:00"},{"alias_kind":"pith_short_8","alias_value":"3ARB77OY","created_at":"2026-07-04T15:37:46.182000+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/3ARB77OY4KCEWFEN3JRRLAUS4U","json":"https://pith.science/pith/3ARB77OY4KCEWFEN3JRRLAUS4U.json","graph_json":"https://pith.science/api/pith-number/3ARB77OY4KCEWFEN3JRRLAUS4U/graph.json","events_json":"https://pith.science/api/pith-number/3ARB77OY4KCEWFEN3JRRLAUS4U/events.json","paper":"https://pith.science/paper/3ARB77OY"},"agent_actions":{"view_html":"https://pith.science/pith/3ARB77OY4KCEWFEN3JRRLAUS4U","download_json":"https://pith.science/pith/3ARB77OY4KCEWFEN3JRRLAUS4U.json","view_paper":"https://pith.science/paper/3ARB77OY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0712.2815&json=true","fetch_graph":"https://pith.science/api/pith-number/3ARB77OY4KCEWFEN3JRRLAUS4U/graph.json","fetch_events":"https://pith.science/api/pith-number/3ARB77OY4KCEWFEN3JRRLAUS4U/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3ARB77OY4KCEWFEN3JRRLAUS4U/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3ARB77OY4KCEWFEN3JRRLAUS4U/action/storage_attestation","attest_author":"https://pith.science/pith/3ARB77OY4KCEWFEN3JRRLAUS4U/action/author_attestation","sign_citation":"https://pith.science/pith/3ARB77OY4KCEWFEN3JRRLAUS4U/action/citation_signature","submit_replication":"https://pith.science/pith/3ARB77OY4KCEWFEN3JRRLAUS4U/action/replication_record"}},"created_at":"2026-07-04T15:37:46.182000+00:00","updated_at":"2026-07-04T15:37:46.182000+00:00"}