{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:2AIMOYAOGXYAPOA3W5G5V6XIV5","short_pith_number":"pith:2AIMOYAO","schema_version":"1.0","canonical_sha256":"d010c7600e35f007b81bb74ddafae8af675e172f07c9aec2e0ebc49678f0a577","source":{"kind":"arxiv","id":"1612.04778","version":1},"attestation_state":"computed","paper":{"title":"A note on the growth of nearly holomorphic vector-valued Siegel modular forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Abhishek Saha, Ameya Pitale, Ralf Schmidt","submitted_at":"2016-12-14T19:27:31Z","abstract_excerpt":"Let $F$ be a nearly holomorphic vector-valued Siegel modular form of weight $\\rho$ with respect to some congruence subgroup of $\\mathrm{Sp}_{2n}(\\mathbb Q)$. In this note, we prove that the function on $\\mathrm{Sp}_{2n}(\\mathbb R)$ obtained by lifting $F$ has the moderate growth (or \"slowly increasing\") property. This is a consequence of the following bound that we prove: $\\|\\rho(Y^{1/2})F(Z) \\| \\ll \\prod_{i=1}^n (\\mu_i(Y)^{\\lambda_1/2} + \\mu_i(Y)^{-\\lambda_1/2})$ where $ \\lambda_1 \\ge \\ldots \\ge \\lambda_n$ is the highest weight of $\\rho$ and $\\mu_i(Y)$ are the eigenvalues of the matrix $Y$."},"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":"1612.04778","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-12-14T19:27:31Z","cross_cats_sorted":[],"title_canon_sha256":"94c8f59a0740497129b70b871ec1e183732307aa78e4784ebb3c6164792cfb2c","abstract_canon_sha256":"eda54a6e8da7471e86c691bb792e5f0a7b34c9fe74252c5b34098e83c879075e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:54:58.639215Z","signature_b64":"KL+aFJTcnKuepz4ZLKYA+X1rsEVlE8tUNBbvWBBzpn4n9IC9GVWydvjNWYHAgFKLkyrALP5TX1vO+ww4D60uDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d010c7600e35f007b81bb74ddafae8af675e172f07c9aec2e0ebc49678f0a577","last_reissued_at":"2026-05-18T00:54:58.638606Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:54:58.638606Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on the growth of nearly holomorphic vector-valued Siegel modular forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Abhishek Saha, Ameya Pitale, Ralf Schmidt","submitted_at":"2016-12-14T19:27:31Z","abstract_excerpt":"Let $F$ be a nearly holomorphic vector-valued Siegel modular form of weight $\\rho$ with respect to some congruence subgroup of $\\mathrm{Sp}_{2n}(\\mathbb Q)$. In this note, we prove that the function on $\\mathrm{Sp}_{2n}(\\mathbb R)$ obtained by lifting $F$ has the moderate growth (or \"slowly increasing\") property. This is a consequence of the following bound that we prove: $\\|\\rho(Y^{1/2})F(Z) \\| \\ll \\prod_{i=1}^n (\\mu_i(Y)^{\\lambda_1/2} + \\mu_i(Y)^{-\\lambda_1/2})$ where $ \\lambda_1 \\ge \\ldots \\ge \\lambda_n$ is the highest weight of $\\rho$ and $\\mu_i(Y)$ are the eigenvalues of the matrix $Y$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.04778","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":"1612.04778","created_at":"2026-05-18T00:54:58.638701+00:00"},{"alias_kind":"arxiv_version","alias_value":"1612.04778v1","created_at":"2026-05-18T00:54:58.638701+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.04778","created_at":"2026-05-18T00:54:58.638701+00:00"},{"alias_kind":"pith_short_12","alias_value":"2AIMOYAOGXYA","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_16","alias_value":"2AIMOYAOGXYAPOA3","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_8","alias_value":"2AIMOYAO","created_at":"2026-05-18T12:29:52.810259+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/2AIMOYAOGXYAPOA3W5G5V6XIV5","json":"https://pith.science/pith/2AIMOYAOGXYAPOA3W5G5V6XIV5.json","graph_json":"https://pith.science/api/pith-number/2AIMOYAOGXYAPOA3W5G5V6XIV5/graph.json","events_json":"https://pith.science/api/pith-number/2AIMOYAOGXYAPOA3W5G5V6XIV5/events.json","paper":"https://pith.science/paper/2AIMOYAO"},"agent_actions":{"view_html":"https://pith.science/pith/2AIMOYAOGXYAPOA3W5G5V6XIV5","download_json":"https://pith.science/pith/2AIMOYAOGXYAPOA3W5G5V6XIV5.json","view_paper":"https://pith.science/paper/2AIMOYAO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1612.04778&json=true","fetch_graph":"https://pith.science/api/pith-number/2AIMOYAOGXYAPOA3W5G5V6XIV5/graph.json","fetch_events":"https://pith.science/api/pith-number/2AIMOYAOGXYAPOA3W5G5V6XIV5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2AIMOYAOGXYAPOA3W5G5V6XIV5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2AIMOYAOGXYAPOA3W5G5V6XIV5/action/storage_attestation","attest_author":"https://pith.science/pith/2AIMOYAOGXYAPOA3W5G5V6XIV5/action/author_attestation","sign_citation":"https://pith.science/pith/2AIMOYAOGXYAPOA3W5G5V6XIV5/action/citation_signature","submit_replication":"https://pith.science/pith/2AIMOYAOGXYAPOA3W5G5V6XIV5/action/replication_record"}},"created_at":"2026-05-18T00:54:58.638701+00:00","updated_at":"2026-05-18T00:54:58.638701+00:00"}