{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:2LO6J6UBSTOQKLL4HRRW7PDSSR","short_pith_number":"pith:2LO6J6UB","canonical_record":{"source":{"id":"0811.1495","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2008-11-10T16:04:47Z","cross_cats_sorted":[],"title_canon_sha256":"811fc8f3f0760e8d8b7926b155ea4bf3a6f5330c90028f34dc887f665c8946af","abstract_canon_sha256":"a4a1d56f7cf7c6e94a29a8ce791998f6bc699a7880b73bb99f211063b8b56aae"},"schema_version":"1.0"},"canonical_sha256":"d2dde4fa8194dd052d7c3c636fbc7294662b22f7bcb804ea7b02108a0e5a4eba","source":{"kind":"arxiv","id":"0811.1495","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0811.1495","created_at":"2026-07-04T15:15:57Z"},{"alias_kind":"arxiv_version","alias_value":"0811.1495v1","created_at":"2026-07-04T15:15:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0811.1495","created_at":"2026-07-04T15:15:57Z"},{"alias_kind":"pith_short_12","alias_value":"2LO6J6UBSTOQ","created_at":"2026-07-04T15:15:57Z"},{"alias_kind":"pith_short_16","alias_value":"2LO6J6UBSTOQKLL4","created_at":"2026-07-04T15:15:57Z"},{"alias_kind":"pith_short_8","alias_value":"2LO6J6UB","created_at":"2026-07-04T15:15:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:2LO6J6UBSTOQKLL4HRRW7PDSSR","target":"record","payload":{"canonical_record":{"source":{"id":"0811.1495","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2008-11-10T16:04:47Z","cross_cats_sorted":[],"title_canon_sha256":"811fc8f3f0760e8d8b7926b155ea4bf3a6f5330c90028f34dc887f665c8946af","abstract_canon_sha256":"a4a1d56f7cf7c6e94a29a8ce791998f6bc699a7880b73bb99f211063b8b56aae"},"schema_version":"1.0"},"canonical_sha256":"d2dde4fa8194dd052d7c3c636fbc7294662b22f7bcb804ea7b02108a0e5a4eba","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:15:57.416623Z","signature_b64":"4osQS/kh0U70thEZAYkm9BNuw+6BSTADu98WRPh/KfeiJg22/PJ88y8DbTAo23Zt4ysGd8S/XMRotLA8dioxBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d2dde4fa8194dd052d7c3c636fbc7294662b22f7bcb804ea7b02108a0e5a4eba","last_reissued_at":"2026-07-04T15:15:57.416273Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:15:57.416273Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0811.1495","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T15:15:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QvKLvP5qdxY7f2Hvtqdf/MU3cVgvDKTcXLiTuAsrXIzi1BtW8oFPYStezYCXHGDvzvzw8p3Wz5D2teOF3BhUCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T08:02:53.461593Z"},"content_sha256":"4a2b29960711395fe01281a23b84817775b48b23a64cc0fc19c12b9f92b585ee","schema_version":"1.0","event_id":"sha256:4a2b29960711395fe01281a23b84817775b48b23a64cc0fc19c12b9f92b585ee"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:2LO6J6UBSTOQKLL4HRRW7PDSSR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the problem of detecting linear dependence 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":"2008-11-10T16:04:47Z","abstract_excerpt":"Let G be the product of an abelian variety and a torus defined over a number field K. Let R be a point in G(K) and let L be a finitely generated subgroup of G(K). Suppose that for all but finitely many primes p of K the point (R mod p) belongs to (L mod p). Does it follow that R belongs to L? We answer this question affirmatively in three cases: if L is cyclic; if L is a free left End_K G-submodule of G(K); if L has a set of generators (as a group) which is a basis of a free left End_K G-submodule of G(K). In general we prove that there exists an integer m (depending only on G, K and the rank "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0811.1495","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/0811.1495/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T15:15:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jkvy5hk5hfYODFIpcYfV+YvTZ6q6QM4e9klqrXwhynvB7d0iLfpY8DyHGbp84UU00LeMDSDW/vwOIPZxBjAFBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T08:02:53.462001Z"},"content_sha256":"f131fc72c864e2ab93d5426902209cd89bb890ed527a423e28ea79743e998440","schema_version":"1.0","event_id":"sha256:f131fc72c864e2ab93d5426902209cd89bb890ed527a423e28ea79743e998440"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2LO6J6UBSTOQKLL4HRRW7PDSSR/bundle.json","state_url":"https://pith.science/pith/2LO6J6UBSTOQKLL4HRRW7PDSSR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2LO6J6UBSTOQKLL4HRRW7PDSSR/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-05T08:02:53Z","links":{"resolver":"https://pith.science/pith/2LO6J6UBSTOQKLL4HRRW7PDSSR","bundle":"https://pith.science/pith/2LO6J6UBSTOQKLL4HRRW7PDSSR/bundle.json","state":"https://pith.science/pith/2LO6J6UBSTOQKLL4HRRW7PDSSR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2LO6J6UBSTOQKLL4HRRW7PDSSR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:2LO6J6UBSTOQKLL4HRRW7PDSSR","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":"a4a1d56f7cf7c6e94a29a8ce791998f6bc699a7880b73bb99f211063b8b56aae","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2008-11-10T16:04:47Z","title_canon_sha256":"811fc8f3f0760e8d8b7926b155ea4bf3a6f5330c90028f34dc887f665c8946af"},"schema_version":"1.0","source":{"id":"0811.1495","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0811.1495","created_at":"2026-07-04T15:15:57Z"},{"alias_kind":"arxiv_version","alias_value":"0811.1495v1","created_at":"2026-07-04T15:15:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0811.1495","created_at":"2026-07-04T15:15:57Z"},{"alias_kind":"pith_short_12","alias_value":"2LO6J6UBSTOQ","created_at":"2026-07-04T15:15:57Z"},{"alias_kind":"pith_short_16","alias_value":"2LO6J6UBSTOQKLL4","created_at":"2026-07-04T15:15:57Z"},{"alias_kind":"pith_short_8","alias_value":"2LO6J6UB","created_at":"2026-07-04T15:15:57Z"}],"graph_snapshots":[{"event_id":"sha256:f131fc72c864e2ab93d5426902209cd89bb890ed527a423e28ea79743e998440","target":"graph","created_at":"2026-07-04T15:15:57Z","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/0811.1495/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 R be a point in G(K) and let L be a finitely generated subgroup of G(K). Suppose that for all but finitely many primes p of K the point (R mod p) belongs to (L mod p). Does it follow that R belongs to L? We answer this question affirmatively in three cases: if L is cyclic; if L is a free left End_K G-submodule of G(K); if L has a set of generators (as a group) which is a basis of a free left End_K G-submodule of G(K). In general we prove that there exists an integer m (depending only on G, K and the rank ","authors_text":"Antonella Perucca","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2008-11-10T16:04:47Z","title":"On the problem of detecting linear dependence for products of abelian varieties and tori"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0811.1495","kind":"arxiv","version":1},"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":"sha256:4a2b29960711395fe01281a23b84817775b48b23a64cc0fc19c12b9f92b585ee","target":"record","created_at":"2026-07-04T15:15:57Z","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":"a4a1d56f7cf7c6e94a29a8ce791998f6bc699a7880b73bb99f211063b8b56aae","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2008-11-10T16:04:47Z","title_canon_sha256":"811fc8f3f0760e8d8b7926b155ea4bf3a6f5330c90028f34dc887f665c8946af"},"schema_version":"1.0","source":{"id":"0811.1495","kind":"arxiv","version":1}},"canonical_sha256":"d2dde4fa8194dd052d7c3c636fbc7294662b22f7bcb804ea7b02108a0e5a4eba","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d2dde4fa8194dd052d7c3c636fbc7294662b22f7bcb804ea7b02108a0e5a4eba","first_computed_at":"2026-07-04T15:15:57.416273Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T15:15:57.416273Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4osQS/kh0U70thEZAYkm9BNuw+6BSTADu98WRPh/KfeiJg22/PJ88y8DbTAo23Zt4ysGd8S/XMRotLA8dioxBQ==","signature_status":"signed_v1","signed_at":"2026-07-04T15:15:57.416623Z","signed_message":"canonical_sha256_bytes"},"source_id":"0811.1495","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4a2b29960711395fe01281a23b84817775b48b23a64cc0fc19c12b9f92b585ee","sha256:f131fc72c864e2ab93d5426902209cd89bb890ed527a423e28ea79743e998440"],"state_sha256":"c86e79d5acf664aaf9474fe5ab3c19168902a25dfd17a8d0b4fc86fd6c2e5e69"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+rWeO3ZJA6TKdk/9nCUBlCw+ugcyQFb5h6k0ajeFmg84Xajmvzw9hl2QFjkJauUDiajL5GE4dx3SH6qmcJJOCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-05T08:02:53.464025Z","bundle_sha256":"fe813a02534d52424780f87cca7a5c9f00717b0963aadc5b285e3d5c7228eff8"}}