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We prove that it is null-homologous and describe actions of various groups on it. For any three normalised cuspidal eigenforms $f_1, f_2, f_3$ of weight $2$ and level $\\Gamma_0(p)$, we prove that the global root number of the twisted triple product $L$-function $L(f_1\\otimes f_2\\otimes f_3\\otimes \\chi, s)$"},"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":"2410.06063","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-10-08T14:10:14Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"df5ecc19a24ad6443cdb2a063231946cc5720a26499c9fe1469e3d38b19d960c","abstract_canon_sha256":"79b75d16e173f08ebef6e16e40a5e4924c308bec7d7f0cbd4e00649d6f8763ee"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-24T01:14:55.787224Z","signature_b64":"89srFZvtpScK/Nihvvs13okzlbWTQjOYLQNZWWUfX+5phhOEkpjU9x/CPlZh/fgvRbap2Fx2AmiEp8stbAobAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"97c649b48b802e3d78704d477d6bfc6068d9a00942b7ac7f36a3eeeca2056b07","last_reissued_at":"2026-06-24T01:14:55.786779Z","signature_status":"signed_v1","first_computed_at":"2026-06-24T01:14:55.786779Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Twisted triple product root numbers and a cycle of Darmon-Rotger","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"David T.-B. 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For any three normalised cuspidal eigenforms $f_1, f_2, f_3$ of weight $2$ and level $\\Gamma_0(p)$, we prove that the global root number of the twisted triple product $L$-function $L(f_1\\otimes f_2\\otimes f_3\\otimes \\chi, s)$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.06063","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/2410.06063/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":"2410.06063","created_at":"2026-06-24T01:14:55.786837+00:00"},{"alias_kind":"arxiv_version","alias_value":"2410.06063v2","created_at":"2026-06-24T01:14:55.786837+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2410.06063","created_at":"2026-06-24T01:14:55.786837+00:00"},{"alias_kind":"pith_short_12","alias_value":"S7DETNELQAXD","created_at":"2026-06-24T01:14:55.786837+00:00"},{"alias_kind":"pith_short_16","alias_value":"S7DETNELQAXD26DQ","created_at":"2026-06-24T01:14:55.786837+00:00"},{"alias_kind":"pith_short_8","alias_value":"S7DETNEL","created_at":"2026-06-24T01:14:55.786837+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/S7DETNELQAXD26DQJVDX2274MB","json":"https://pith.science/pith/S7DETNELQAXD26DQJVDX2274MB.json","graph_json":"https://pith.science/api/pith-number/S7DETNELQAXD26DQJVDX2274MB/graph.json","events_json":"https://pith.science/api/pith-number/S7DETNELQAXD26DQJVDX2274MB/events.json","paper":"https://pith.science/paper/S7DETNEL"},"agent_actions":{"view_html":"https://pith.science/pith/S7DETNELQAXD26DQJVDX2274MB","download_json":"https://pith.science/pith/S7DETNELQAXD26DQJVDX2274MB.json","view_paper":"https://pith.science/paper/S7DETNEL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2410.06063&json=true","fetch_graph":"https://pith.science/api/pith-number/S7DETNELQAXD26DQJVDX2274MB/graph.json","fetch_events":"https://pith.science/api/pith-number/S7DETNELQAXD26DQJVDX2274MB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/S7DETNELQAXD26DQJVDX2274MB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/S7DETNELQAXD26DQJVDX2274MB/action/storage_attestation","attest_author":"https://pith.science/pith/S7DETNELQAXD26DQJVDX2274MB/action/author_attestation","sign_citation":"https://pith.science/pith/S7DETNELQAXD26DQJVDX2274MB/action/citation_signature","submit_replication":"https://pith.science/pith/S7DETNELQAXD26DQJVDX2274MB/action/replication_record"}},"created_at":"2026-06-24T01:14:55.786837+00:00","updated_at":"2026-06-24T01:14:55.786837+00:00"}