{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:3ARB77OY4KCEWFEN3JRRLAUS4U","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"2e8126d4e7f9b8462045d3c3e81a9816fafef69960d54511a17f628844c4ef76","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2007-12-17T20:58:32Z","title_canon_sha256":"f02d71509d26a86ce79f109792283b1758e7f1281edc3fa2522d61fe28909126"},"schema_version":"1.0","source":{"id":"0712.2815","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0712.2815","created_at":"2026-07-04T15:37:46Z"},{"alias_kind":"arxiv_version","alias_value":"0712.2815v3","created_at":"2026-07-04T15:37:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0712.2815","created_at":"2026-07-04T15:37:46Z"},{"alias_kind":"pith_short_12","alias_value":"3ARB77OY4KCE","created_at":"2026-07-04T15:37:46Z"},{"alias_kind":"pith_short_16","alias_value":"3ARB77OY4KCEWFEN","created_at":"2026-07-04T15:37:46Z"},{"alias_kind":"pith_short_8","alias_value":"3ARB77OY","created_at":"2026-07-04T15:37:46Z"}],"graph_snapshots":[{"event_id":"sha256:71e57ae712b7731292954b68f5ccf5fe880d9d3025875cccac6c448451228972","target":"graph","created_at":"2026-07-04T15:37:46Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/0712.2815/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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).","authors_text":"Antonella Perucca","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2007-12-17T20:58:32Z","title":"Two variants of the support problem for products of abelian varieties and tori"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0712.2815","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:d851aab31354a5323e8fc2ab6491c682f99b070ecde4fde0d0b1f9a17c83478e","target":"record","created_at":"2026-07-04T15:37:46Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"2e8126d4e7f9b8462045d3c3e81a9816fafef69960d54511a17f628844c4ef76","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2007-12-17T20:58:32Z","title_canon_sha256":"f02d71509d26a86ce79f109792283b1758e7f1281edc3fa2522d61fe28909126"},"schema_version":"1.0","source":{"id":"0712.2815","kind":"arxiv","version":3}},"canonical_sha256":"d8221ffdd8e2844b148dda63158292e532e43c33d723969bf85ba1f4f7532305","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d8221ffdd8e2844b148dda63158292e532e43c33d723969bf85ba1f4f7532305","first_computed_at":"2026-07-04T15:37:46.181934Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T15:37:46.181934Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uEbxt8KoqI5k8edZ4oTwMe3AANfJzvhM0Sxbn+orSFkElwaAhLKNVGAlAH1/C5kuKfyKS1wM9CeGx1GzzgshCQ==","signature_status":"signed_v1","signed_at":"2026-07-04T15:37:46.182385Z","signed_message":"canonical_sha256_bytes"},"source_id":"0712.2815","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d851aab31354a5323e8fc2ab6491c682f99b070ecde4fde0d0b1f9a17c83478e","sha256:71e57ae712b7731292954b68f5ccf5fe880d9d3025875cccac6c448451228972"],"state_sha256":"4f61ab7d38d1abb27545bee61755ac19c9ffe3ddb5abe69f7057b06647252ea1"}