{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:GDUGJMDEHJL7NY25THBIAY4OTW","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":"3c9bf1f7f9a3e18bfa74ba5ea94c5ccfaa07a98f89f6667b8dcfb780b5228996","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-06-23T06:33:08Z","title_canon_sha256":"9739bd483869fa447c75b62bc0a384ef6d03703d4874df20d638dcef55766851"},"schema_version":"1.0","source":{"id":"1406.5815","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1406.5815","created_at":"2026-05-18T02:49:10Z"},{"alias_kind":"arxiv_version","alias_value":"1406.5815v1","created_at":"2026-05-18T02:49:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.5815","created_at":"2026-05-18T02:49:10Z"},{"alias_kind":"pith_short_12","alias_value":"GDUGJMDEHJL7","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_16","alias_value":"GDUGJMDEHJL7NY25","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_8","alias_value":"GDUGJMDE","created_at":"2026-05-18T12:28:30Z"}],"graph_snapshots":[{"event_id":"sha256:b3a613da08ce59e1764efc906f0c84d2076c8f6d3b63624ee8d9d7d3e3fa5d50","target":"graph","created_at":"2026-05-18T02:49:10Z","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 prove a functional equation for two projective systems of finite abelian $p$-groups, $\\{\\fa_n\\}$ and $\\{\\fb_n\\}$, endowed with an action of $\\ZZ_p^d$ such that $\\fa_n$ can be identified with the Pontryagin dual of $\\fb_n$ for all $n$.\n  Let $K$ be a global field. Let $L$ be a $\\ZZ_p^d$-extension of $K$ ($d\\geq 1$), unramified outside a finite set of places. Let $A$ be an abelian variety over $K$. We prove an algebraic functional equation for the Pontryagin dual of the Selmer group of $A$.","authors_text":"Fabien Trihan, Ignazio Longhi, King Fai Lai, Ki-Seng Tan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-06-23T06:33:08Z","title":"Pontryagin duality for Iwasawa modules and abelian varieties"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.5815","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":"sha256:7da7e626a43056d040d8a80e1648f11b383f6ea4709d227c7ae58c9c63aee548","target":"record","created_at":"2026-05-18T02:49:10Z","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":"3c9bf1f7f9a3e18bfa74ba5ea94c5ccfaa07a98f89f6667b8dcfb780b5228996","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-06-23T06:33:08Z","title_canon_sha256":"9739bd483869fa447c75b62bc0a384ef6d03703d4874df20d638dcef55766851"},"schema_version":"1.0","source":{"id":"1406.5815","kind":"arxiv","version":1}},"canonical_sha256":"30e864b0643a57f6e35d99c280638e9daccb76d9630856b4e1995e0dacf0b73f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"30e864b0643a57f6e35d99c280638e9daccb76d9630856b4e1995e0dacf0b73f","first_computed_at":"2026-05-18T02:49:10.700972Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:49:10.700972Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"V7gw2QzdrMm0XG9Ae0yqRpsIj+mxnQlVyjhvOIjrEMrx4FieQcFvdoUzvTZ3o5PgV8qy7wnHwQg38esOlgAACA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:49:10.701583Z","signed_message":"canonical_sha256_bytes"},"source_id":"1406.5815","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7da7e626a43056d040d8a80e1648f11b383f6ea4709d227c7ae58c9c63aee548","sha256:b3a613da08ce59e1764efc906f0c84d2076c8f6d3b63624ee8d9d7d3e3fa5d50"],"state_sha256":"ab3ddc73dac364922649873292737c65d81e833e564aa4ef96238ab49af71ace"}