{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:RGDSNPDDZ7QC4V5YOZYD77NFSO","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"6c4b1159c29573a52002f43e292cbb0fba2a29e69b10930fa8c143f8ec73e9bd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-03-15T02:52:53Z","title_canon_sha256":"f81e1ed326c3c3f914f1d79e4ad09f089fabcb5f8a48a8c18cffb80472ecf7bd"},"schema_version":"1.0","source":{"id":"1403.3748","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1403.3748","created_at":"2026-05-18T02:26:52Z"},{"alias_kind":"arxiv_version","alias_value":"1403.3748v1","created_at":"2026-05-18T02:26:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.3748","created_at":"2026-05-18T02:26:52Z"},{"alias_kind":"pith_short_12","alias_value":"RGDSNPDDZ7QC","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_16","alias_value":"RGDSNPDDZ7QC4V5Y","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_8","alias_value":"RGDSNPDD","created_at":"2026-05-18T12:28:46Z"}],"graph_snapshots":[{"event_id":"sha256:339a76a16995fbb04addb88aaca4809a43e6cf37df6ebf534cb2a9d87bafb51d","target":"graph","created_at":"2026-05-18T02:26:52Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We are interested in the harmonic analysis on $p$-adic homogeneous spaces based on spherical functions. In the present paper, we investigate the space $X$ of unitary hermitian matrices of odd size over a ${\\mathfrak p}$-adic field of odd residual characteristic, which is a continuation of our previous paper where we have studied for even size matrices. First we give the explicit representatives of the Cartan decomposition of $X$ and introduce a typical spherical function $\\omega(x;z)$ on $X$. After studying the functional equations, we give an explicit formula for $\\omega(x;z)$, where Hall-Lit","authors_text":"Yasushi Komori, Yumiko Hironaka","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-03-15T02:52:53Z","title":"Spherical functions on the space of $p$-adic unitary hermitian matrices II, the case of odd size"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.3748","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6912498385c4b1bd1f15cce3a4057aa4e4b78b6653e1b322a2f1eed7be0cdd3e","target":"record","created_at":"2026-05-18T02:26:52Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"6c4b1159c29573a52002f43e292cbb0fba2a29e69b10930fa8c143f8ec73e9bd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-03-15T02:52:53Z","title_canon_sha256":"f81e1ed326c3c3f914f1d79e4ad09f089fabcb5f8a48a8c18cffb80472ecf7bd"},"schema_version":"1.0","source":{"id":"1403.3748","kind":"arxiv","version":1}},"canonical_sha256":"898726bc63cfe02e57b876703ffda59389c019d182a63f34c00a3fd36e443101","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"898726bc63cfe02e57b876703ffda59389c019d182a63f34c00a3fd36e443101","first_computed_at":"2026-05-18T02:26:52.956300Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:26:52.956300Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yJPS1IFBCieKxDO8RSd3/GPc7fYz8Ew8R63Qnr8w9oPOAB23mcqhwWhxtR8A6NyuOtDPsIurxF7fz2Sn+TP0BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:26:52.956714Z","signed_message":"canonical_sha256_bytes"},"source_id":"1403.3748","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6912498385c4b1bd1f15cce3a4057aa4e4b78b6653e1b322a2f1eed7be0cdd3e","sha256:339a76a16995fbb04addb88aaca4809a43e6cf37df6ebf534cb2a9d87bafb51d"],"state_sha256":"1a7002f308b7ecb021977b39e3e3f067142f047ce21d9bb53d3ff3f65b9ba145"}