{"paper":{"title":"Mod $\\ell$ gamma factors and a converse theorem for finite general linear groups","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.NT","authors_text":"Ashwin Iyengar, Gilbert Moss, Heidi Goodson, Jacksyn Bakeberg, Mathilde Gerbelli-Gauthier, Robin Zhang","submitted_at":"2023-07-14T19:33:37Z","abstract_excerpt":"The local converse theorem for Rankin-Selberg gamma factors of $\\mathrm{GL}_2(\\mathbb{F}_q)$ proved by Piatetski-Shapiro over $\\mathbb{C}$ no longer holds after reduction modulo $\\ell \\neq p$. To remedy this, we construct new $\\mathrm{GL}_n \\times \\mathrm{GL}_m$ gamma factors valued in arbitrary $\\mathbb{Z}[1/p, \\zeta_p]$-algebras for Whittaker-type representations, show that they satisfy a functional equation, and then prove a $\\mathrm{GL}_n \\times \\mathrm{GL}_{n-1}$ converse theorem for irreducible cuspidal representations. In the $\\mathrm{GL}_2 \\times \\mathrm{GL}_1$ case, we define an alter"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.07593","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2307.07593/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}