{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:DM5EPN4JGLSB4Z5YPAVZNNS5A5","short_pith_number":"pith:DM5EPN4J","schema_version":"1.0","canonical_sha256":"1b3a47b78932e41e67b8782b96b65d07597d1918caf7dca86eebce13b69154c5","source":{"kind":"arxiv","id":"2605.16803","version":1},"attestation_state":"computed","paper":{"title":"Explicit Formulas for the Casimir Eigenvalues of $SL(n,\\mathbb{Z})$-Maass Forms","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Maass forms for SL(n,Z) have their Casimir eigenvalues given explicitly by formulas in the Langlands parameters.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Vishal Muthuvel","submitted_at":"2026-05-16T04:15:50Z","abstract_excerpt":"Maass forms for $SL(n,\\mathbb{Z})$ are defined to be eigenfunctions of the Casimir operators $\\mathcal{D}_{m,n}$ of orders $1 \\leq m \\leq n$ for $GL(n,\\mathbb{R})$. For any $1 \\leq m \\leq n$ and Maass form $\\phi$ for $SL(n,\\mathbb{Z})$, we provide a formula for the eigenvalue of $\\mathcal{D}_{m,n}$ associated with $\\phi$ in terms of the Langlands parameters of $\\phi$. In the case $m=2$, we recover the formula for the Laplace eigenvalue of a Maass form due to Terras, the Casimir differential operator of order $2$ being the Laplacian. Our proof takes a graph-theoretic approach, relating the acti"},"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":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2605.16803","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-16T04:15:50Z","cross_cats_sorted":[],"title_canon_sha256":"a02cd7c334b8664ee936cc70c21e7811717286a80c5a944e2b1214054ea65be4","abstract_canon_sha256":"f3244df05aca0af754224ca731421784d0e91c9a133cfcdf8e4c8dea61baefcc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:03:23.007627Z","signature_b64":"6i3KuV+StpEvjUhJim1J61PCvx3QzSPQ/D3NKQLNFbNsjaBK0PbZzlmI/GcZ2QnWarj1OjdEFi6BM1VtmO3TAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1b3a47b78932e41e67b8782b96b65d07597d1918caf7dca86eebce13b69154c5","last_reissued_at":"2026-05-20T00:03:23.006773Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:03:23.006773Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Explicit Formulas for the Casimir Eigenvalues of $SL(n,\\mathbb{Z})$-Maass Forms","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Maass forms for SL(n,Z) have their Casimir eigenvalues given explicitly by formulas in the Langlands parameters.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Vishal Muthuvel","submitted_at":"2026-05-16T04:15:50Z","abstract_excerpt":"Maass forms for $SL(n,\\mathbb{Z})$ are defined to be eigenfunctions of the Casimir operators $\\mathcal{D}_{m,n}$ of orders $1 \\leq m \\leq n$ for $GL(n,\\mathbb{R})$. For any $1 \\leq m \\leq n$ and Maass form $\\phi$ for $SL(n,\\mathbb{Z})$, we provide a formula for the eigenvalue of $\\mathcal{D}_{m,n}$ associated with $\\phi$ in terms of the Langlands parameters of $\\phi$. In the case $m=2$, we recover the formula for the Laplace eigenvalue of a Maass form due to Terras, the Casimir differential operator of order $2$ being the Laplacian. Our proof takes a graph-theoretic approach, relating the acti"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For any 1 ≤ m ≤ n and Maass form φ for SL(n,Z), we provide a formula for the eigenvalue of D_{m,n} associated with φ in terms of the Langlands parameters of φ.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The Maass forms for SL(n,Z) are eigenfunctions of the full set of Casimir operators D_{m,n} of orders 1 ≤ m ≤ n for GL(n,R), and the Langlands parameters are the standard ones that classify these forms.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Explicit formulas are derived for the eigenvalues of Casimir operators D_{m,n} on SL(n,Z)-Maass forms in terms of Langlands parameters, with a graph-theoretic proof that recovers the known Laplacian case for m=2.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Maass forms for SL(n,Z) have their Casimir eigenvalues given explicitly by formulas in the Langlands parameters.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"9e93f2cb75fac98c9aacb4eca1b374af3da20b45ee56e8c3778074b86eed7b04"},"source":{"id":"2605.16803","kind":"arxiv","version":1},"verdict":{"id":"22780a9a-a32f-4d27-b568-94996ce8f048","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T20:06:29.048917Z","strongest_claim":"For any 1 ≤ m ≤ n and Maass form φ for SL(n,Z), we provide a formula for the eigenvalue of D_{m,n} associated with φ in terms of the Langlands parameters of φ.","one_line_summary":"Explicit formulas are derived for the eigenvalues of Casimir operators D_{m,n} on SL(n,Z)-Maass forms in terms of Langlands parameters, with a graph-theoretic proof that recovers the known Laplacian case for m=2.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The Maass forms for SL(n,Z) are eigenfunctions of the full set of Casimir operators D_{m,n} of orders 1 ≤ m ≤ n for GL(n,R), and the Langlands parameters are the standard ones that classify these forms.","pith_extraction_headline":"Maass forms for SL(n,Z) have their Casimir eigenvalues given explicitly by formulas in the Langlands parameters."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16803/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T20:31:19.133757Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T20:22:04.485470Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T19:01:56.285403Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T18:33:26.422297Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"49f137d150365d6b881fc6ebe76b04892d17d0f0b61b3ccedf969d2130a3587e"},"references":{"count":3,"sample":[{"doi":"","year":1979,"title":"Eisenstein Series and the Trace Formula","work_id":"9ffc8dcc-bbac-4e64-bf21-dbd95e3b38f3","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1951,"title":"On some applications of the universal en- veloping algebra of a semisimple Lie algebra","work_id":"704b5d8b-2d1d-4562-bb18-15337e55884d","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1956,"title":"Harmonic Analysis and Discontinuous Groups in Weakly Symmetric Riemannian Spaces With Applications to Dirichlet Series","work_id":"b2699b1d-b56c-499e-bc8c-51b87fd213ad","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":3,"snapshot_sha256":"50dd6410de4bc3990e9abc7cffc99b0bd0583d1a67bedc0042d6cfe34a37c358","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"d1accc0b76fe2ce88a281830003d2211215c3cc89ac7c3ac17c599cf78262501"},"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":"2605.16803","created_at":"2026-05-20T00:03:23.006915+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.16803v1","created_at":"2026-05-20T00:03:23.006915+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.16803","created_at":"2026-05-20T00:03:23.006915+00:00"},{"alias_kind":"pith_short_12","alias_value":"DM5EPN4JGLSB","created_at":"2026-05-20T00:03:23.006915+00:00"},{"alias_kind":"pith_short_16","alias_value":"DM5EPN4JGLSB4Z5Y","created_at":"2026-05-20T00:03:23.006915+00:00"},{"alias_kind":"pith_short_8","alias_value":"DM5EPN4J","created_at":"2026-05-20T00:03:23.006915+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":2,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DM5EPN4JGLSB4Z5YPAVZNNS5A5","json":"https://pith.science/pith/DM5EPN4JGLSB4Z5YPAVZNNS5A5.json","graph_json":"https://pith.science/api/pith-number/DM5EPN4JGLSB4Z5YPAVZNNS5A5/graph.json","events_json":"https://pith.science/api/pith-number/DM5EPN4JGLSB4Z5YPAVZNNS5A5/events.json","paper":"https://pith.science/paper/DM5EPN4J"},"agent_actions":{"view_html":"https://pith.science/pith/DM5EPN4JGLSB4Z5YPAVZNNS5A5","download_json":"https://pith.science/pith/DM5EPN4JGLSB4Z5YPAVZNNS5A5.json","view_paper":"https://pith.science/paper/DM5EPN4J","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.16803&json=true","fetch_graph":"https://pith.science/api/pith-number/DM5EPN4JGLSB4Z5YPAVZNNS5A5/graph.json","fetch_events":"https://pith.science/api/pith-number/DM5EPN4JGLSB4Z5YPAVZNNS5A5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DM5EPN4JGLSB4Z5YPAVZNNS5A5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DM5EPN4JGLSB4Z5YPAVZNNS5A5/action/storage_attestation","attest_author":"https://pith.science/pith/DM5EPN4JGLSB4Z5YPAVZNNS5A5/action/author_attestation","sign_citation":"https://pith.science/pith/DM5EPN4JGLSB4Z5YPAVZNNS5A5/action/citation_signature","submit_replication":"https://pith.science/pith/DM5EPN4JGLSB4Z5YPAVZNNS5A5/action/replication_record"}},"created_at":"2026-05-20T00:03:23.006915+00:00","updated_at":"2026-05-20T00:03:23.006915+00:00"}