{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:ECM7CWLQS4NO357NXBEHOYOCBL","short_pith_number":"pith:ECM7CWLQ","canonical_record":{"source":{"id":"1709.09847","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-09-28T08:40:52Z","cross_cats_sorted":["math.AC","math.AG"],"title_canon_sha256":"2d7d9f63c075e25adce3b64d257463382f3d7368b334e97369529bd11ffcc096","abstract_canon_sha256":"626548f041c8f5e5596c90f68580717c0c4a10da2d51f7a7ca76efc436409a89"},"schema_version":"1.0"},"canonical_sha256":"2099f15970971aedf7edb8487761c20ae3e8631d29191a655a25a3f3529edb03","source":{"kind":"arxiv","id":"1709.09847","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.09847","created_at":"2026-05-18T00:34:06Z"},{"alias_kind":"arxiv_version","alias_value":"1709.09847v1","created_at":"2026-05-18T00:34:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.09847","created_at":"2026-05-18T00:34:06Z"},{"alias_kind":"pith_short_12","alias_value":"ECM7CWLQS4NO","created_at":"2026-05-18T12:31:12Z"},{"alias_kind":"pith_short_16","alias_value":"ECM7CWLQS4NO357N","created_at":"2026-05-18T12:31:12Z"},{"alias_kind":"pith_short_8","alias_value":"ECM7CWLQ","created_at":"2026-05-18T12:31:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:ECM7CWLQS4NO357NXBEHOYOCBL","target":"record","payload":{"canonical_record":{"source":{"id":"1709.09847","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-09-28T08:40:52Z","cross_cats_sorted":["math.AC","math.AG"],"title_canon_sha256":"2d7d9f63c075e25adce3b64d257463382f3d7368b334e97369529bd11ffcc096","abstract_canon_sha256":"626548f041c8f5e5596c90f68580717c0c4a10da2d51f7a7ca76efc436409a89"},"schema_version":"1.0"},"canonical_sha256":"2099f15970971aedf7edb8487761c20ae3e8631d29191a655a25a3f3529edb03","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:34:06.524584Z","signature_b64":"oDWa1kIl0C8kA1WxjDMnH3Da90LGfhX6gP4RgqXerumxL7aXkFuPxxsl3V9pywgvnbaeYnlrXZgtAFfHzhnPCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2099f15970971aedf7edb8487761c20ae3e8631d29191a655a25a3f3529edb03","last_reissued_at":"2026-05-18T00:34:06.524028Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:34:06.524028Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1709.09847","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":"2026-05-18T00:34:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VPk6R+5XgTKz6lZiLURKTVwLCbnzZ8FcjJo9i5Iq7ESuSWi1YxMXeoeoX2AYNCBiryhli7njaHRhC3aPvWkyBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T15:20:25.513956Z"},"content_sha256":"00f5f7a3d22cd64fba8d897b7a2949db97b5d9e0f8c171f67f70f4d6bdd1b8c0","schema_version":"1.0","event_id":"sha256:00f5f7a3d22cd64fba8d897b7a2949db97b5d9e0f8c171f67f70f4d6bdd1b8c0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:ECM7CWLQS4NO357NXBEHOYOCBL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Dual pairs of algebras and finite commutative group schemes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.AG"],"primary_cat":"math.NT","authors_text":"Peter Bruin","submitted_at":"2017-09-28T08:40:52Z","abstract_excerpt":"We introduce a category of dual pairs of finite locally free algebras over a ring. This gives an efficient way to represent finite locally free commutative group schemes. We give a number of algorithms to compute with dual pairs of algebras, and we apply our results to Galois representations on finite Abelian groups."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.09847","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":""},"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":"2026-05-18T00:34:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8mM7x6dLNP7cTiBNWoYW62a1Sgrc5+LbKknzH5umsjU8hhSOH+gPft0pV3QnriXfjbnL1eC6f2GYD0mFilSLCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T15:20:25.514448Z"},"content_sha256":"3f484db30d44115c040d28d2ef564d856acfb25288b294632f132e44411ee4cd","schema_version":"1.0","event_id":"sha256:3f484db30d44115c040d28d2ef564d856acfb25288b294632f132e44411ee4cd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ECM7CWLQS4NO357NXBEHOYOCBL/bundle.json","state_url":"https://pith.science/pith/ECM7CWLQS4NO357NXBEHOYOCBL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ECM7CWLQS4NO357NXBEHOYOCBL/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":"2026-05-27T15:20:25Z","links":{"resolver":"https://pith.science/pith/ECM7CWLQS4NO357NXBEHOYOCBL","bundle":"https://pith.science/pith/ECM7CWLQS4NO357NXBEHOYOCBL/bundle.json","state":"https://pith.science/pith/ECM7CWLQS4NO357NXBEHOYOCBL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ECM7CWLQS4NO357NXBEHOYOCBL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:ECM7CWLQS4NO357NXBEHOYOCBL","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":"626548f041c8f5e5596c90f68580717c0c4a10da2d51f7a7ca76efc436409a89","cross_cats_sorted":["math.AC","math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-09-28T08:40:52Z","title_canon_sha256":"2d7d9f63c075e25adce3b64d257463382f3d7368b334e97369529bd11ffcc096"},"schema_version":"1.0","source":{"id":"1709.09847","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.09847","created_at":"2026-05-18T00:34:06Z"},{"alias_kind":"arxiv_version","alias_value":"1709.09847v1","created_at":"2026-05-18T00:34:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.09847","created_at":"2026-05-18T00:34:06Z"},{"alias_kind":"pith_short_12","alias_value":"ECM7CWLQS4NO","created_at":"2026-05-18T12:31:12Z"},{"alias_kind":"pith_short_16","alias_value":"ECM7CWLQS4NO357N","created_at":"2026-05-18T12:31:12Z"},{"alias_kind":"pith_short_8","alias_value":"ECM7CWLQ","created_at":"2026-05-18T12:31:12Z"}],"graph_snapshots":[{"event_id":"sha256:3f484db30d44115c040d28d2ef564d856acfb25288b294632f132e44411ee4cd","target":"graph","created_at":"2026-05-18T00:34:06Z","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"},"paper":{"abstract_excerpt":"We introduce a category of dual pairs of finite locally free algebras over a ring. This gives an efficient way to represent finite locally free commutative group schemes. We give a number of algorithms to compute with dual pairs of algebras, and we apply our results to Galois representations on finite Abelian groups.","authors_text":"Peter Bruin","cross_cats":["math.AC","math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-09-28T08:40:52Z","title":"Dual pairs of algebras and finite commutative group schemes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.09847","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":"sha256:00f5f7a3d22cd64fba8d897b7a2949db97b5d9e0f8c171f67f70f4d6bdd1b8c0","target":"record","created_at":"2026-05-18T00:34:06Z","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":"626548f041c8f5e5596c90f68580717c0c4a10da2d51f7a7ca76efc436409a89","cross_cats_sorted":["math.AC","math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-09-28T08:40:52Z","title_canon_sha256":"2d7d9f63c075e25adce3b64d257463382f3d7368b334e97369529bd11ffcc096"},"schema_version":"1.0","source":{"id":"1709.09847","kind":"arxiv","version":1}},"canonical_sha256":"2099f15970971aedf7edb8487761c20ae3e8631d29191a655a25a3f3529edb03","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2099f15970971aedf7edb8487761c20ae3e8631d29191a655a25a3f3529edb03","first_computed_at":"2026-05-18T00:34:06.524028Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:34:06.524028Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"oDWa1kIl0C8kA1WxjDMnH3Da90LGfhX6gP4RgqXerumxL7aXkFuPxxsl3V9pywgvnbaeYnlrXZgtAFfHzhnPCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:34:06.524584Z","signed_message":"canonical_sha256_bytes"},"source_id":"1709.09847","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:00f5f7a3d22cd64fba8d897b7a2949db97b5d9e0f8c171f67f70f4d6bdd1b8c0","sha256:3f484db30d44115c040d28d2ef564d856acfb25288b294632f132e44411ee4cd"],"state_sha256":"129e0efaf751d5484bd5f2a1102f564d71190731eae3cf4d24f74d4134325160"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yfxo9MYpuZ2eGEU3uiFgv0KfbzuHSYFAO8hND9HAMxe0M5NBke8m3LmgGuVjAv7aN6nP/XWGSwFJimfnxzBNCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T15:20:25.517982Z","bundle_sha256":"fcfc53b4884cf32c30a24eb7297784c2fdbd2326e7e34ba247f7173e9e742430"}}