{"paper":{"title":"Lifting Subgroups of Symplectic Groups over $\\mathbb{Z} / \\ell \\mathbb{Z}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT","math.RT"],"primary_cat":"math.GR","authors_text":"Aaron Landesman, Ashvin Swaminathan, James Tao, Yujie Xu","submitted_at":"2016-07-16T02:20:13Z","abstract_excerpt":"For a positive integer $g$, let $\\mathrm{Sp}_{2g}(R)$ denote the group of $2g \\times 2g$ symplectic matrices over a ring $R$. Assume $g \\ge 2$. For a prime number $\\ell$, we give a self-contained proof that any closed subgroup of $\\mathrm{Sp}_{2g}(\\mathbb{Z}_\\ell)$ which surjects onto $\\mathrm{Sp}_{2g}(\\mathbb{Z}/\\ell\\mathbb{Z})$ must in fact equal all of $\\mathrm{Sp}_{2g}(\\mathbb{Z}_\\ell)$. The result and the method of proof are both motivated by group-theoretic considerations that arise in the study of Galois representations associated to abelian varieties."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.04698","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":""},"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"}