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Let $\\underline{III}(E/F)$ denote the Tate-Shafarevich group of $E$ over $F$ and $ L(E/F,s) $ be the Hasse-Weil complex $L$-function of $E$ over $F$. Under some technical assumptions, we show that when $rank_{\\mathbb{Z}} \\hspace{0.01mm} \\hspace{1mm} E(F) = 1$ and $\\#\\Big(\\underline{III}(E/F)_ {p^\\infty}\\Big) < \\infty$, then $ord_{s=1} \\ L(E/F,s) = 1$. Further, we give an"},"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":"2504.21799","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-04-30T16:56:10Z","cross_cats_sorted":[],"title_canon_sha256":"b28e1ed2bb128f20687a8206c1d0ba0dbc0dc95646243ce886e10bd779e47faf","abstract_canon_sha256":"2266c70df5d08a147722db0c14c53fffaffd45b6d70b33398e81d8eb9fa4fd94"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-24T01:14:20.160794Z","signature_b64":"eTVFwEEIuxLxQ1YW5EEUX01LxArXI0EdjLMFEVnyyKv9IAQYAq5CNmn7o0DEvsYptliWyVYQGvZKFGbpGN0UDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8fc6736c73075f10435ff53ea0d5a1d53beab61947fc88c172fef21375632427","last_reissued_at":"2026-06-24T01:14:20.160278Z","signature_status":"signed_v1","first_computed_at":"2026-06-24T01:14:20.160278Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A $p$-Converse theorem for Real Quadratic Fields","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Aprameyo Pal, Guhan Venkat, Muskan Bansal, Somnath Jha","submitted_at":"2025-04-30T16:56:10Z","abstract_excerpt":"Let $E$ be an elliptic curve defined over a real quadratic field $F$. Let $p > 5$ be a rational prime that is inert in $F$ and assume that $E$ has split multiplicative reduction at the prime $\\mathfrak{p}$ of $F$ dividing $p$. Let $\\underline{III}(E/F)$ denote the Tate-Shafarevich group of $E$ over $F$ and $ L(E/F,s) $ be the Hasse-Weil complex $L$-function of $E$ over $F$. Under some technical assumptions, we show that when $rank_{\\mathbb{Z}} \\hspace{0.01mm} \\hspace{1mm} E(F) = 1$ and $\\#\\Big(\\underline{III}(E/F)_ {p^\\infty}\\Big) < \\infty$, then $ord_{s=1} \\ L(E/F,s) = 1$. Further, we give an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.21799","kind":"arxiv","version":3},"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/2504.21799/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":"2504.21799","created_at":"2026-06-24T01:14:20.160341+00:00"},{"alias_kind":"arxiv_version","alias_value":"2504.21799v3","created_at":"2026-06-24T01:14:20.160341+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.21799","created_at":"2026-06-24T01:14:20.160341+00:00"},{"alias_kind":"pith_short_12","alias_value":"R7DHG3DTA5PR","created_at":"2026-06-24T01:14:20.160341+00:00"},{"alias_kind":"pith_short_16","alias_value":"R7DHG3DTA5PRAQ27","created_at":"2026-06-24T01:14:20.160341+00:00"},{"alias_kind":"pith_short_8","alias_value":"R7DHG3DT","created_at":"2026-06-24T01:14:20.160341+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/R7DHG3DTA5PRAQ276U7KBVNB2U","json":"https://pith.science/pith/R7DHG3DTA5PRAQ276U7KBVNB2U.json","graph_json":"https://pith.science/api/pith-number/R7DHG3DTA5PRAQ276U7KBVNB2U/graph.json","events_json":"https://pith.science/api/pith-number/R7DHG3DTA5PRAQ276U7KBVNB2U/events.json","paper":"https://pith.science/paper/R7DHG3DT"},"agent_actions":{"view_html":"https://pith.science/pith/R7DHG3DTA5PRAQ276U7KBVNB2U","download_json":"https://pith.science/pith/R7DHG3DTA5PRAQ276U7KBVNB2U.json","view_paper":"https://pith.science/paper/R7DHG3DT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2504.21799&json=true","fetch_graph":"https://pith.science/api/pith-number/R7DHG3DTA5PRAQ276U7KBVNB2U/graph.json","fetch_events":"https://pith.science/api/pith-number/R7DHG3DTA5PRAQ276U7KBVNB2U/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/R7DHG3DTA5PRAQ276U7KBVNB2U/action/timestamp_anchor","attest_storage":"https://pith.science/pith/R7DHG3DTA5PRAQ276U7KBVNB2U/action/storage_attestation","attest_author":"https://pith.science/pith/R7DHG3DTA5PRAQ276U7KBVNB2U/action/author_attestation","sign_citation":"https://pith.science/pith/R7DHG3DTA5PRAQ276U7KBVNB2U/action/citation_signature","submit_replication":"https://pith.science/pith/R7DHG3DTA5PRAQ276U7KBVNB2U/action/replication_record"}},"created_at":"2026-06-24T01:14:20.160341+00:00","updated_at":"2026-06-24T01:14:20.160341+00:00"}