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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 $A$."},"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":"1406.5815","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-06-23T06:33:08Z","cross_cats_sorted":[],"title_canon_sha256":"9739bd483869fa447c75b62bc0a384ef6d03703d4874df20d638dcef55766851","abstract_canon_sha256":"3c9bf1f7f9a3e18bfa74ba5ea94c5ccfaa07a98f89f6667b8dcfb780b5228996"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:49:10.701583Z","signature_b64":"V7gw2QzdrMm0XG9Ae0yqRpsIj+mxnQlVyjhvOIjrEMrx4FieQcFvdoUzvTZ3o5PgV8qy7wnHwQg38esOlgAACA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"30e864b0643a57f6e35d99c280638e9daccb76d9630856b4e1995e0dacf0b73f","last_reissued_at":"2026-05-18T02:49:10.700972Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:49:10.700972Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Pontryagin duality for Iwasawa modules and abelian varieties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Fabien Trihan, Ignazio Longhi, King Fai Lai, Ki-Seng Tan","submitted_at":"2014-06-23T06:33:08Z","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 $A$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.5815","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1406.5815","created_at":"2026-05-18T02:49:10.701044+00:00"},{"alias_kind":"arxiv_version","alias_value":"1406.5815v1","created_at":"2026-05-18T02:49:10.701044+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.5815","created_at":"2026-05-18T02:49:10.701044+00:00"},{"alias_kind":"pith_short_12","alias_value":"GDUGJMDEHJL7","created_at":"2026-05-18T12:28:30.664211+00:00"},{"alias_kind":"pith_short_16","alias_value":"GDUGJMDEHJL7NY25","created_at":"2026-05-18T12:28:30.664211+00:00"},{"alias_kind":"pith_short_8","alias_value":"GDUGJMDE","created_at":"2026-05-18T12:28:30.664211+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GDUGJMDEHJL7NY25THBIAY4OTW","json":"https://pith.science/pith/GDUGJMDEHJL7NY25THBIAY4OTW.json","graph_json":"https://pith.science/api/pith-number/GDUGJMDEHJL7NY25THBIAY4OTW/graph.json","events_json":"https://pith.science/api/pith-number/GDUGJMDEHJL7NY25THBIAY4OTW/events.json","paper":"https://pith.science/paper/GDUGJMDE"},"agent_actions":{"view_html":"https://pith.science/pith/GDUGJMDEHJL7NY25THBIAY4OTW","download_json":"https://pith.science/pith/GDUGJMDEHJL7NY25THBIAY4OTW.json","view_paper":"https://pith.science/paper/GDUGJMDE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1406.5815&json=true","fetch_graph":"https://pith.science/api/pith-number/GDUGJMDEHJL7NY25THBIAY4OTW/graph.json","fetch_events":"https://pith.science/api/pith-number/GDUGJMDEHJL7NY25THBIAY4OTW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GDUGJMDEHJL7NY25THBIAY4OTW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GDUGJMDEHJL7NY25THBIAY4OTW/action/storage_attestation","attest_author":"https://pith.science/pith/GDUGJMDEHJL7NY25THBIAY4OTW/action/author_attestation","sign_citation":"https://pith.science/pith/GDUGJMDEHJL7NY25THBIAY4OTW/action/citation_signature","submit_replication":"https://pith.science/pith/GDUGJMDEHJL7NY25THBIAY4OTW/action/replication_record"}},"created_at":"2026-05-18T02:49:10.701044+00:00","updated_at":"2026-05-18T02:49:10.701044+00:00"}