{"paper":{"title":"Hecke Eigenvalues of Ikeda Lifts","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Hecke eigenvalues of Ikeda lifts are positive for all sufficiently large primes.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ameya Pitale, Nagarjuna Chary Addanki","submitted_at":"2026-05-15T15:45:08Z","abstract_excerpt":"In this paper, we study the Hecke eigenvalues of Ikeda lifts. Using the spherical map for the Hecke algebra of the symplectic group, we obtain an explicit formula for the eigenvalues $\\lambda_F(p^r)$. From this formula, we show that $\\lambda_F(p^r)$ can be written as a polynomial in $p^{\\pm 1/2}$ with a positive leading term. Furthermore, we prove that the coefficients of this polynomial are bounded and, as a consequence, the Hecke eigenvalues $\\lambda_F(p^r)$ are positive for all sufficiently large primes $p$."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"λ_F(p^r) can be written as a polynomial in p^{±1/2} with a positive leading term; the coefficients of this polynomial are bounded, and therefore λ_F(p^r) is positive for all sufficiently large primes p.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The spherical map for the Hecke algebra of the symplectic group correctly transfers the Hecke eigenvalues from the Ikeda lift to an explicit algebraic expression that can be analyzed as a polynomial.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Derives explicit formula for Hecke eigenvalues of Ikeda lifts as polynomials in p^{±1/2} with bounded coefficients and proves positivity for large primes.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Hecke eigenvalues of Ikeda lifts are positive for all sufficiently large primes.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"116f71c2bec7531a7826f3745464b83b2d90558baa0f383eb8873d2f8e806809"},"source":{"id":"2605.16083","kind":"arxiv","version":1},"verdict":{"id":"4a4883d1-3175-4631-b604-0153fb46186c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T18:47:08.804843Z","strongest_claim":"λ_F(p^r) can be written as a polynomial in p^{±1/2} with a positive leading term; the coefficients of this polynomial are bounded, and therefore λ_F(p^r) is positive for all sufficiently large primes p.","one_line_summary":"Derives explicit formula for Hecke eigenvalues of Ikeda lifts as polynomials in p^{±1/2} with bounded coefficients and proves positivity for large primes.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The spherical map for the Hecke algebra of the symplectic group correctly transfers the Hecke eigenvalues from the Ikeda lift to an explicit algebraic expression that can be analyzed as a polynomial.","pith_extraction_headline":"Hecke eigenvalues of Ikeda lifts are positive for all sufficiently large primes."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16083/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T19:01:32.128843Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T19:01:18.969763Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T17:33:41.535591Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T16:41:55.502841Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"04c95475a40cfda903865224a6235190807cb6102ed4a9d2313162b2867bd0a5"},"references":{"count":17,"sample":[{"doi":"","year":2024,"title":"N. C. Addanki. On signs of eigenvalues of Siegel modular forms satisfying Ramanujan conjec- ture.arXiv, 2024","work_id":"339a6301-3d8b-4ae4-9b88-306f3857d90f","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"N. C. Addanki. On signs of Hecke eigenvalues of Ikeda lifts.Ramanujan J., 66(4):81, 2025","work_id":"68f0baf9-92a5-4c57-ab38-aa1c6e019331","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2009,"title":"Andrianov.Introduction to Siegel modular forms and Dirichlet series","work_id":"9eb421d3-10bc-4778-82e2-887f6a3914ac","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1970,"title":"A. N. Andrianov. Spherical functions forGLn over local fields, and the summation of Hecke series.Mat. Sb. (N.S.), 83(125):429–451, 1970","work_id":"74f6ef9f-621e-4b43-b7ad-5468b08c5c44","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1974,"title":"A. N. Andrianov. Euler products that correspond to Siegel’s modular forms of genus2.Uspehi Mat. Nauk, 29(3(177)):43–110, 1974","work_id":"27d9a231-bbe6-4cf5-ba23-30491caa8b3f","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":17,"snapshot_sha256":"d3e0f03f2fa28277b9e00abd2c518db25fdbc860bb005986ec0f5c577251eee7","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"a9867439d027754b390156d8f5f2bccd40ba038de27a3fa48bae200658440a95"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}