{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:HPJWFZKSRESDBYZM5WPIUBFGRK","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":"161ca944c0ef4d90ccd214ec2599c0838fecc9f5505bc441af1c4648c6bbb5df","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-12-19T14:10:09Z","title_canon_sha256":"750ab1dd497193d68cbc1cbaccbd48b8331913affd42336b2828c11f8865cc08"},"schema_version":"1.0","source":{"id":"1612.06187","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.06187","created_at":"2026-05-18T00:54:42Z"},{"alias_kind":"arxiv_version","alias_value":"1612.06187v1","created_at":"2026-05-18T00:54:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.06187","created_at":"2026-05-18T00:54:42Z"},{"alias_kind":"pith_short_12","alias_value":"HPJWFZKSRESD","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"HPJWFZKSRESDBYZM","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"HPJWFZKS","created_at":"2026-05-18T12:30:19Z"}],"graph_snapshots":[{"event_id":"sha256:ed149f7855677d00f6a8e56df3a6c78e7ba193571197b32f3cc833017d42b0c2","target":"graph","created_at":"2026-05-18T00:54:42Z","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":"Mazur and Rubin have recently developed a theory of higher rank Kolyvagin and Stark systems over principal artinian rings and discrete valuation rings. In this article we describe a natural extension of (a slightly modified version of) their theory to systems over more general coefficient rings. We also construct unconditionally, and for general $p$-adic representations, a canonical, and typically large, module of higher rank Euler systems and show that for $p$-adic representations satisfying standard hypotheses the image under a natural higher rank Kolyvagin-derivative type homomorphism of ea","authors_text":"David Burns, Takamichi Sano","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-12-19T14:10:09Z","title":"On the theory of higher rank Euler, Kolyvagin and Stark systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.06187","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":"sha256:1d2dd5847350bead08efc4331b588797abed2deba2714a7ebc0c728548a727e6","target":"record","created_at":"2026-05-18T00:54:42Z","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":"161ca944c0ef4d90ccd214ec2599c0838fecc9f5505bc441af1c4648c6bbb5df","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-12-19T14:10:09Z","title_canon_sha256":"750ab1dd497193d68cbc1cbaccbd48b8331913affd42336b2828c11f8865cc08"},"schema_version":"1.0","source":{"id":"1612.06187","kind":"arxiv","version":1}},"canonical_sha256":"3bd362e552892430e32ced9e8a04a68aac7f84f2d9d8cff7390dc719dd4501ba","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3bd362e552892430e32ced9e8a04a68aac7f84f2d9d8cff7390dc719dd4501ba","first_computed_at":"2026-05-18T00:54:42.944341Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:54:42.944341Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yhUPAABFQEbt5wJjeVXr3rLVxGIhemLG6gBAb21fPToPqjmbAS7wWkcC27Tg+BkHmBeTclQfWWpTEXT6yCxGDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:54:42.944703Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.06187","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1d2dd5847350bead08efc4331b588797abed2deba2714a7ebc0c728548a727e6","sha256:ed149f7855677d00f6a8e56df3a6c78e7ba193571197b32f3cc833017d42b0c2"],"state_sha256":"99db30086d85cb41b635d9d3233495f3b16a5f1a41f7aaf23f74dea2c225c2d1"}