{"paper":{"title":"Transport of Zariski density in compatible collections of $G$-representations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jake Huryn, Yifei Zhang","submitted_at":"2024-11-13T00:18:47Z","abstract_excerpt":"Let $X$ be a connected normal scheme of finite type over $\\mathbf{Z}$, let $G$ be a connected reductive group over $\\mathbf{Q}$, and let $\\{\\rho_\\ell\\colon\\pi_1(X[1/\\ell])\\to G(\\mathbf{Q}_\\ell)\\}_\\ell$ be a Frobenius-compatible collection of continuous homomorphisms indexed by the primes. Assume $\\mathrm{Img}(\\rho_\\ell)$ is Zariski-dense in $G_{\\mathbf{Q}_\\ell}$ for all $\\ell$ in a nonempty finite set $\\mathcal{R}$. We prove that, under certain hypotheses on $\\mathcal{R}$ (depending only on $G$), $\\mathrm{Img}(\\rho_\\ell)$ is Zariski-dense in $G_{\\mathbf{Q}_\\ell}$ for all $\\ell$ in a set of Dir"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.08259","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.08259/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"}