{"paper":{"title":"Joint Sato-Tate Laws for Transformations of Hecke Eigenvalues: The Vertical Case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A framework combining a higher-dimensional Erdős-Turán analogue with Hardy-Krause variation approximations delivers effective joint Sato-Tate laws for Hecke eigenvalues and Frobenius traces.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Mohammad H. Hamdar, Tian Wang","submitted_at":"2026-04-27T17:55:14Z","abstract_excerpt":"We introduce a framework within which a large class of joint equidistribution problems can be studied and resolved with effective error terms. This involves proving a higher dimensional and $\\mu$-analogue of the Erd\\\"{o}s-Tur\\'{a}n inequality, and utilizing the theory of the Hardy-Krause (H-K) variation from analysis, where, in particular, we formulate a technique to approximate a broad class of relevant functions by functions of bounded H-K variation. Our main focus will be on the vertical Sato-Tate problem for spaces of cusp forms and for families of elliptic curves over finite fields. In pa"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We obtain novel results concerning the distribution of arithmetic relations, and, more generally, multi-dimensional functions of Fourier coefficients and Frobenius traces.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That a broad class of relevant functions can be approximated by functions of bounded Hardy-Krause variation and that the higher-dimensional μ-analogue of the Erdős-Turán inequality applies with effective error terms to the vertical Sato-Tate problems.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A new analytic framework establishes joint Sato-Tate laws with effective errors for transformations of Hecke eigenvalues in the vertical case for cusp forms and elliptic curves.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A framework combining a higher-dimensional Erdős-Turán analogue with Hardy-Krause variation approximations delivers effective joint Sato-Tate laws for Hecke eigenvalues and Frobenius traces.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"6ff2821d63c6f05845b4539a9653981c6adbd8d1e14407fd8d82c4613d81eb05"},"source":{"id":"2604.24753","kind":"arxiv","version":2},"verdict":{"id":"c5fa66d1-839d-4610-b0a3-9944b2f403b8","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T01:27:14.719276Z","strongest_claim":"We obtain novel results concerning the distribution of arithmetic relations, and, more generally, multi-dimensional functions of Fourier coefficients and Frobenius traces.","one_line_summary":"A new analytic framework establishes joint Sato-Tate laws with effective errors for transformations of Hecke eigenvalues in the vertical case for cusp forms and elliptic curves.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That a broad class of relevant functions can be approximated by functions of bounded Hardy-Krause variation and that the higher-dimensional μ-analogue of the Erdős-Turán inequality applies with effective error terms to the vertical Sato-Tate problems.","pith_extraction_headline":"A framework combining a higher-dimensional Erdős-Turán analogue with Hardy-Krause variation approximations delivers effective joint Sato-Tate laws for Hecke eigenvalues and Frobenius traces."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.24753/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T06:34:30.729673Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T21:44:41.624501Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"1b6c9f22cd6c737c12c7e93e07cbe2014d24f8dc521d87a4c8e22de12ab33f9a"},"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"}