{"paper":{"title":"An approximation principle for congruence subgroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Erez Lapid, Tobias Finis","submitted_at":"2013-08-16T11:16:10Z","abstract_excerpt":"The motivating question of this paper is roughly the following: given a group scheme $G$ over $\\mathbb{Z}_p$, $p$ prime, with semisimple generic fiber $G_{\\mathbb{Q}_p}$, how far are open subgroups of $G(\\mathbb{Z}_p)$ from subgroups of the form $X(\\mathbb{Z}_p)\\mathbf{K}_p(p^n)$, where $X$ is a subgroup scheme of $G$ and $\\mathbf{K}_p(p^n)$ is the principal congruence subgroup $\\operatorname{Ker} (G(\\mathbb{Z}_p)\\rightarrow G(\\mathbb{Z}/p^n\\mathbb{Z}))$? More precisely, we will show that for $G_{\\mathbb{Q}_p}$ simply connected there exist constants $J\\ge1$ and $\\varepsilon>0$, depending only "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.3604","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}