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Using a $p$-integral version of Rallis inner product formula and modularity theorems for $\\mathrm{GSp}_{4/\\mathbb{Q}}$ and $\\mathrm{U}_{4/\\mathbb{Q}}$, we establish an identity between the $p$-part of the critical value at $1$ of the degree $5$ $L$-function of $\\pi$ twisted by the non-trivial quadratic Dirichlet character $\\xi$ associated to the"},"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":"1811.02031","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-11-05T20:55:44Z","cross_cats_sorted":[],"title_canon_sha256":"7c8d98fa64011d35e166e10979fd0b2f1d586e250c5a8d6aab38b10eb458b5f2","abstract_canon_sha256":"5de0a1e26cdcdc8fb652cd5b86d9d83848147880c9e87a0a3c4bc426cab75eb1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:01:32.516097Z","signature_b64":"P0faa4ZuSA+U9hY/N+rrQySbVhlKRrbYo0fwBKiXkyc8E7eiaE7du7lgJOqDUaAimzlX3toR/lC2PcWqvHYLBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0629193cf8bad00687771867282cc0f6b38021d0907098b0b8531045e1bf491d","last_reissued_at":"2026-05-18T00:01:32.515684Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:01:32.515684Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Special L-values and Selmer groups of Siegel modular forms of genus 2","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Xiaoyu Zhang","submitted_at":"2018-11-05T20:55:44Z","abstract_excerpt":"Let $p$ be an odd prime, $N$ a square-free odd positive integer prime to $p$, $\\pi$ a $p$-ordinary cohomological irreducible cuspidal automorphic representation of $\\mathrm{GSp}_4(\\mathbb{A}_\\mathbb{Q})$ of principal level $N$ and Iwahori level at $p$. Using a $p$-integral version of Rallis inner product formula and modularity theorems for $\\mathrm{GSp}_{4/\\mathbb{Q}}$ and $\\mathrm{U}_{4/\\mathbb{Q}}$, we establish an identity between the $p$-part of the critical value at $1$ of the degree $5$ $L$-function of $\\pi$ twisted by the non-trivial quadratic Dirichlet character $\\xi$ associated to the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.02031","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":"1811.02031","created_at":"2026-05-18T00:01:32.515749+00:00"},{"alias_kind":"arxiv_version","alias_value":"1811.02031v1","created_at":"2026-05-18T00:01:32.515749+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.02031","created_at":"2026-05-18T00:01:32.515749+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/AYURSPHYXLIANB3XDBTSQLGA62","json":"https://pith.science/pith/AYURSPHYXLIANB3XDBTSQLGA62.json","graph_json":"https://pith.science/api/pith-number/AYURSPHYXLIANB3XDBTSQLGA62/graph.json","events_json":"https://pith.science/api/pith-number/AYURSPHYXLIANB3XDBTSQLGA62/events.json","paper":"https://pith.science/paper/AYURSPHY"},"agent_actions":{"view_html":"https://pith.science/pith/AYURSPHYXLIANB3XDBTSQLGA62","download_json":"https://pith.science/pith/AYURSPHYXLIANB3XDBTSQLGA62.json","view_paper":"https://pith.science/paper/AYURSPHY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1811.02031&json=true","fetch_graph":"https://pith.science/api/pith-number/AYURSPHYXLIANB3XDBTSQLGA62/graph.json","fetch_events":"https://pith.science/api/pith-number/AYURSPHYXLIANB3XDBTSQLGA62/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AYURSPHYXLIANB3XDBTSQLGA62/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AYURSPHYXLIANB3XDBTSQLGA62/action/storage_attestation","attest_author":"https://pith.science/pith/AYURSPHYXLIANB3XDBTSQLGA62/action/author_attestation","sign_citation":"https://pith.science/pith/AYURSPHYXLIANB3XDBTSQLGA62/action/citation_signature","submit_replication":"https://pith.science/pith/AYURSPHYXLIANB3XDBTSQLGA62/action/replication_record"}},"created_at":"2026-05-18T00:01:32.515749+00:00","updated_at":"2026-05-18T00:01:32.515749+00:00"}