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We prove the hybrid subconvexity bounds for the twisted $L$-functions \\[ L(1/2,\\phi\\times u_j\\times\\chi)\\ll_{\\phi,\\varepsilon} (qt_j)^{3/2-\\theta+\\varepsilon},\\quad L(1/2+it,\\phi\\times\\chi)\\ll_{\\phi,\\varepsilon} (qt)^{3/4-\\theta/2+\\varepsilon}, \\] for any $\\varepsilon>0$, where $\\theta=1/23$ is admissible."},"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":"1605.09487","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-05-31T04:07:01Z","cross_cats_sorted":[],"title_canon_sha256":"fab87655f8e1c22f97b94818b71bf4037b1372be21de02b31791352165e3a721","abstract_canon_sha256":"8e41adbadc9e95fe87795ece2027d0540448ef538b43ad8b449b0c85bbb4d813"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:00:33.007503Z","signature_b64":"J9W1bVEJw00Zuaz4/Br4qnCuI5Ot2QcPnOKFL70cI5cWgAm6giUMzbkKbZPQ01Z97dgwNWcB+/EWgLGWDrXOCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8523121b9e0980cc4dd53bf7471ff1136420389d134b460fb5144096285a2421","last_reissued_at":"2026-05-18T00:00:33.006726Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:00:33.006726Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hybrid subconvexity bounds for twisted $L$-functions on $GL(3)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Bingrong Huang","submitted_at":"2016-05-31T04:07:01Z","abstract_excerpt":"Let $q$ be a large prime, and $\\chi$ the quadratic character modulo $q$. Let $\\phi$ be a self-dual Hecke--Maass cusp form for $SL(3,\\mathbb{Z})$, and $u_j$ a Hecke--Maass cusp form for $\\Gamma_0(q)\\subseteq SL(2,\\mathbb{Z})$ with spectral parameter $t_j$. We prove the hybrid subconvexity bounds for the twisted $L$-functions \\[ L(1/2,\\phi\\times u_j\\times\\chi)\\ll_{\\phi,\\varepsilon} (qt_j)^{3/2-\\theta+\\varepsilon},\\quad L(1/2+it,\\phi\\times\\chi)\\ll_{\\phi,\\varepsilon} (qt)^{3/4-\\theta/2+\\varepsilon}, \\] for any $\\varepsilon>0$, where $\\theta=1/23$ is admissible."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.09487","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":""},"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":"1605.09487","created_at":"2026-05-18T00:00:33.006969+00:00"},{"alias_kind":"arxiv_version","alias_value":"1605.09487v2","created_at":"2026-05-18T00:00:33.006969+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.09487","created_at":"2026-05-18T00:00:33.006969+00:00"},{"alias_kind":"pith_short_12","alias_value":"QURREG46BGAM","created_at":"2026-05-18T12:30:41.710351+00:00"},{"alias_kind":"pith_short_16","alias_value":"QURREG46BGAMYTOV","created_at":"2026-05-18T12:30:41.710351+00:00"},{"alias_kind":"pith_short_8","alias_value":"QURREG46","created_at":"2026-05-18T12:30:41.710351+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/QURREG46BGAMYTOVHP3UOH7RCN","json":"https://pith.science/pith/QURREG46BGAMYTOVHP3UOH7RCN.json","graph_json":"https://pith.science/api/pith-number/QURREG46BGAMYTOVHP3UOH7RCN/graph.json","events_json":"https://pith.science/api/pith-number/QURREG46BGAMYTOVHP3UOH7RCN/events.json","paper":"https://pith.science/paper/QURREG46"},"agent_actions":{"view_html":"https://pith.science/pith/QURREG46BGAMYTOVHP3UOH7RCN","download_json":"https://pith.science/pith/QURREG46BGAMYTOVHP3UOH7RCN.json","view_paper":"https://pith.science/paper/QURREG46","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1605.09487&json=true","fetch_graph":"https://pith.science/api/pith-number/QURREG46BGAMYTOVHP3UOH7RCN/graph.json","fetch_events":"https://pith.science/api/pith-number/QURREG46BGAMYTOVHP3UOH7RCN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QURREG46BGAMYTOVHP3UOH7RCN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QURREG46BGAMYTOVHP3UOH7RCN/action/storage_attestation","attest_author":"https://pith.science/pith/QURREG46BGAMYTOVHP3UOH7RCN/action/author_attestation","sign_citation":"https://pith.science/pith/QURREG46BGAMYTOVHP3UOH7RCN/action/citation_signature","submit_replication":"https://pith.science/pith/QURREG46BGAMYTOVHP3UOH7RCN/action/replication_record"}},"created_at":"2026-05-18T00:00:33.006969+00:00","updated_at":"2026-05-18T00:00:33.006969+00:00"}