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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"},"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":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2411.08259","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-11-13T00:18:47Z","cross_cats_sorted":[],"title_canon_sha256":"f9d3c25a413dd3245144777e152ed36e01e2e59867391d391193a05b04e49b0e","abstract_canon_sha256":"7fb389f3ab5a68ce5cbd947cc4d9d0d5a07a315e6a6f26663ee575c4e07be98d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:34:44.257716Z","signature_b64":"us13OeXQNjn4Hd6173s5IC1SNbd1mhpa9ASaqGI1MGWuWUUJjWhtKjpRxPDOM8tHOLeU+0Mp7MD3WIKKGP3sAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"43fc2bd6c5a8ae0ec2c80de17e036cdfe2a76eed67b9ff25dbdc54d02f46d083","last_reissued_at":"2026-07-05T09:34:44.257222Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:34:44.257222Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2411.08259","created_at":"2026-07-05T09:34:44.257281+00:00"},{"alias_kind":"arxiv_version","alias_value":"2411.08259v1","created_at":"2026-07-05T09:34:44.257281+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.08259","created_at":"2026-07-05T09:34:44.257281+00:00"},{"alias_kind":"pith_short_12","alias_value":"IP6CXVWFVCXA","created_at":"2026-07-05T09:34:44.257281+00:00"},{"alias_kind":"pith_short_16","alias_value":"IP6CXVWFVCXA5QWI","created_at":"2026-07-05T09:34:44.257281+00:00"},{"alias_kind":"pith_short_8","alias_value":"IP6CXVWF","created_at":"2026-07-05T09:34:44.257281+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/IP6CXVWFVCXA5QWIBXQX4A3M37","json":"https://pith.science/pith/IP6CXVWFVCXA5QWIBXQX4A3M37.json","graph_json":"https://pith.science/api/pith-number/IP6CXVWFVCXA5QWIBXQX4A3M37/graph.json","events_json":"https://pith.science/api/pith-number/IP6CXVWFVCXA5QWIBXQX4A3M37/events.json","paper":"https://pith.science/paper/IP6CXVWF"},"agent_actions":{"view_html":"https://pith.science/pith/IP6CXVWFVCXA5QWIBXQX4A3M37","download_json":"https://pith.science/pith/IP6CXVWFVCXA5QWIBXQX4A3M37.json","view_paper":"https://pith.science/paper/IP6CXVWF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2411.08259&json=true","fetch_graph":"https://pith.science/api/pith-number/IP6CXVWFVCXA5QWIBXQX4A3M37/graph.json","fetch_events":"https://pith.science/api/pith-number/IP6CXVWFVCXA5QWIBXQX4A3M37/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IP6CXVWFVCXA5QWIBXQX4A3M37/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IP6CXVWFVCXA5QWIBXQX4A3M37/action/storage_attestation","attest_author":"https://pith.science/pith/IP6CXVWFVCXA5QWIBXQX4A3M37/action/author_attestation","sign_citation":"https://pith.science/pith/IP6CXVWFVCXA5QWIBXQX4A3M37/action/citation_signature","submit_replication":"https://pith.science/pith/IP6CXVWFVCXA5QWIBXQX4A3M37/action/replication_record"}},"created_at":"2026-07-05T09:34:44.257281+00:00","updated_at":"2026-07-05T09:34:44.257281+00:00"}