{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2023:RJ7NQJB2TUGLJJNCXQSPYF3RLS","short_pith_number":"pith:RJ7NQJB2","schema_version":"1.0","canonical_sha256":"8a7ed8243a9d0cb4a5a2bc24fc17715cb7b1717b661d86d8840ae2045d30c4e5","source":{"kind":"arxiv","id":"2309.16446","version":2},"attestation_state":"computed","paper":{"title":"Crystalline representations and Wach modules in the relative case II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Abhinandan","submitted_at":"2023-09-28T13:56:36Z","abstract_excerpt":"We study relative Wach modules generalising our previous works on this subject. Our main result shows a categorical equivalence between relative Wach modules and lattices inside relative crystalline representations. Using this result, we deduce a purity statement for relative crystalline representations and provide a criteria for checking crystallinity of relative $p$-adic representations. Furthermore, we interpret relative Wach modules as modules with $q$-connections, and show that for a crystalline representation, its associated Wach module together with the Nygaard filtration is the canonic"},"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":"2309.16446","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2023-09-28T13:56:36Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"30297a2aa127fbe5bedd513c6451498e3064f2a037784345255ed5e2cb29e120","abstract_canon_sha256":"2a6097e8200ab414c3fb9857d3b7ef052370424263927fcbdcc7a7b90a32dba4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:11:07.774211Z","signature_b64":"dvMwBUVnkYHkQB01G+A+R+Fw1P1wX0m2QULLEGCaV6Tb3Uf1qQhwYwnI6FlBM6uKZ2jPCrE+1HRO1bAjLVcDAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8a7ed8243a9d0cb4a5a2bc24fc17715cb7b1717b661d86d8840ae2045d30c4e5","last_reissued_at":"2026-06-19T16:11:07.773863Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:11:07.773863Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Crystalline representations and Wach modules in the relative case II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Abhinandan","submitted_at":"2023-09-28T13:56:36Z","abstract_excerpt":"We study relative Wach modules generalising our previous works on this subject. Our main result shows a categorical equivalence between relative Wach modules and lattices inside relative crystalline representations. Using this result, we deduce a purity statement for relative crystalline representations and provide a criteria for checking crystallinity of relative $p$-adic representations. Furthermore, we interpret relative Wach modules as modules with $q$-connections, and show that for a crystalline representation, its associated Wach module together with the Nygaard filtration is the canonic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.16446","kind":"arxiv","version":2},"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/2309.16446/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":"2309.16446","created_at":"2026-06-19T16:11:07.773918+00:00"},{"alias_kind":"arxiv_version","alias_value":"2309.16446v2","created_at":"2026-06-19T16:11:07.773918+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2309.16446","created_at":"2026-06-19T16:11:07.773918+00:00"},{"alias_kind":"pith_short_12","alias_value":"RJ7NQJB2TUGL","created_at":"2026-06-19T16:11:07.773918+00:00"},{"alias_kind":"pith_short_16","alias_value":"RJ7NQJB2TUGLJJNC","created_at":"2026-06-19T16:11:07.773918+00:00"},{"alias_kind":"pith_short_8","alias_value":"RJ7NQJB2","created_at":"2026-06-19T16:11:07.773918+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2405.18245","citing_title":"Prismatic $F$-crystals and Wach modules","ref_index":1,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/RJ7NQJB2TUGLJJNCXQSPYF3RLS","json":"https://pith.science/pith/RJ7NQJB2TUGLJJNCXQSPYF3RLS.json","graph_json":"https://pith.science/api/pith-number/RJ7NQJB2TUGLJJNCXQSPYF3RLS/graph.json","events_json":"https://pith.science/api/pith-number/RJ7NQJB2TUGLJJNCXQSPYF3RLS/events.json","paper":"https://pith.science/paper/RJ7NQJB2"},"agent_actions":{"view_html":"https://pith.science/pith/RJ7NQJB2TUGLJJNCXQSPYF3RLS","download_json":"https://pith.science/pith/RJ7NQJB2TUGLJJNCXQSPYF3RLS.json","view_paper":"https://pith.science/paper/RJ7NQJB2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2309.16446&json=true","fetch_graph":"https://pith.science/api/pith-number/RJ7NQJB2TUGLJJNCXQSPYF3RLS/graph.json","fetch_events":"https://pith.science/api/pith-number/RJ7NQJB2TUGLJJNCXQSPYF3RLS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RJ7NQJB2TUGLJJNCXQSPYF3RLS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RJ7NQJB2TUGLJJNCXQSPYF3RLS/action/storage_attestation","attest_author":"https://pith.science/pith/RJ7NQJB2TUGLJJNCXQSPYF3RLS/action/author_attestation","sign_citation":"https://pith.science/pith/RJ7NQJB2TUGLJJNCXQSPYF3RLS/action/citation_signature","submit_replication":"https://pith.science/pith/RJ7NQJB2TUGLJJNCXQSPYF3RLS/action/replication_record"}},"created_at":"2026-06-19T16:11:07.773918+00:00","updated_at":"2026-06-19T16:11:07.773918+00:00"}