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The level $\\ell$ theta function $\\Th_{\\L_{\\ell} (C)} $ of $C$ is defined on the lattice $\\L_{\\ell} (C):= \\set {x \\in \\O_K^n : \\rho_\\ell (x) \\in C}$, where $\\rho_{\\ell}:\\O_K \\rightarrow \\O_K/2\\O_K$ is the natural projection. In this paper, we prove that: %\ni) for any $\\ell, \\ell^\\prime$ such that $\\ell \\leq \\ell^\\prime$, $\\Th_{\\Lambda_\\ell}(q)$ and $\\Th_{\\Lambda_{\\ell^\\prime}}(q)$ ha"},"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":"1209.0469","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-09-03T20:00:40Z","cross_cats_sorted":[],"title_canon_sha256":"d0fa0c0494cfedeef27b4b426bfc79101558601046a48081874801f4891d8838","abstract_canon_sha256":"6c2cc52e6704c00eaed990275e65862dc6ecacdda4c32c03a8e999607482e121"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:46:18.043031Z","signature_b64":"w/WgCcOPEtfQ+le1KD3DjPC8smfZrz5OHxWjkRc/OcQxWhOE+WtYeXVKDG5MplID++BhLaylv9fkgfzIccl8Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b53052c1829a00c4ed01c1b3091e378a7600183c55199ed80df4c08f12d7be9a","last_reissued_at":"2026-05-18T03:46:18.042567Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:46:18.042567Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Codes over rings of size four, Hermitian lattices, and corresponding theta functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"G. S. Wijesiri, T. Shaska","submitted_at":"2012-09-03T20:00:40Z","abstract_excerpt":"Let $K=Q(\\sqrt{-\\ell})$ be an imaginary quadratic field with ring of integers $\\O_K$, where $\\ell$ is a square free integer such that $\\ell\\equiv 3 \\mod 4$ and $C=[n, k]$ be a linear code defined over $\\O_K/2\\O_K$. The level $\\ell$ theta function $\\Th_{\\L_{\\ell} (C)} $ of $C$ is defined on the lattice $\\L_{\\ell} (C):= \\set {x \\in \\O_K^n : \\rho_\\ell (x) \\in C}$, where $\\rho_{\\ell}:\\O_K \\rightarrow \\O_K/2\\O_K$ is the natural projection. In this paper, we prove that: %\ni) for any $\\ell, \\ell^\\prime$ such that $\\ell \\leq \\ell^\\prime$, $\\Th_{\\Lambda_\\ell}(q)$ and $\\Th_{\\Lambda_{\\ell^\\prime}}(q)$ ha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.0469","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":"1209.0469","created_at":"2026-05-18T03:46:18.042630+00:00"},{"alias_kind":"arxiv_version","alias_value":"1209.0469v1","created_at":"2026-05-18T03:46:18.042630+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.0469","created_at":"2026-05-18T03:46:18.042630+00:00"},{"alias_kind":"pith_short_12","alias_value":"WUYFFQMCTIAM","created_at":"2026-05-18T12:27:27.928770+00:00"},{"alias_kind":"pith_short_16","alias_value":"WUYFFQMCTIAMJ3IB","created_at":"2026-05-18T12:27:27.928770+00:00"},{"alias_kind":"pith_short_8","alias_value":"WUYFFQMC","created_at":"2026-05-18T12:27:27.928770+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/WUYFFQMCTIAMJ3IBYGZQSHRXRJ","json":"https://pith.science/pith/WUYFFQMCTIAMJ3IBYGZQSHRXRJ.json","graph_json":"https://pith.science/api/pith-number/WUYFFQMCTIAMJ3IBYGZQSHRXRJ/graph.json","events_json":"https://pith.science/api/pith-number/WUYFFQMCTIAMJ3IBYGZQSHRXRJ/events.json","paper":"https://pith.science/paper/WUYFFQMC"},"agent_actions":{"view_html":"https://pith.science/pith/WUYFFQMCTIAMJ3IBYGZQSHRXRJ","download_json":"https://pith.science/pith/WUYFFQMCTIAMJ3IBYGZQSHRXRJ.json","view_paper":"https://pith.science/paper/WUYFFQMC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1209.0469&json=true","fetch_graph":"https://pith.science/api/pith-number/WUYFFQMCTIAMJ3IBYGZQSHRXRJ/graph.json","fetch_events":"https://pith.science/api/pith-number/WUYFFQMCTIAMJ3IBYGZQSHRXRJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WUYFFQMCTIAMJ3IBYGZQSHRXRJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WUYFFQMCTIAMJ3IBYGZQSHRXRJ/action/storage_attestation","attest_author":"https://pith.science/pith/WUYFFQMCTIAMJ3IBYGZQSHRXRJ/action/author_attestation","sign_citation":"https://pith.science/pith/WUYFFQMCTIAMJ3IBYGZQSHRXRJ/action/citation_signature","submit_replication":"https://pith.science/pith/WUYFFQMCTIAMJ3IBYGZQSHRXRJ/action/replication_record"}},"created_at":"2026-05-18T03:46:18.042630+00:00","updated_at":"2026-05-18T03:46:18.042630+00:00"}