{"paper":{"title":"$m$-bigness in compatible systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Paul-James White","submitted_at":"2010-07-02T13:05:44Z","abstract_excerpt":"Taylor-Wiles type lifting theorems allow one to deduce that for $\\rho$ a \"sufficiently nice\" $l$-adic representation of the absolute Galois group of a number field whose semi-simplified reduction modulo $l$, denoted $\\overline{\\rho}$, comes from an automorphic representation then so does $\\rho$. The recent lifting theorems of Barnet-Lamb-Gee-Geraghty-Taylor impose a technical condition, called \\emph{$m$-big}, upon the residual representation $\\overline{\\rho}$. Snowden-Wiles proved that for a sufficiently irreducible compatible system of Galois representations, the residual images are \\emph{big"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.0358","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"}