{"paper":{"title":"Higher congruence companion forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jayanta Manoharmayum, Rajender Adibhatla","submitted_at":"2010-07-01T14:34:27Z","abstract_excerpt":"For a rational prime $p \\geq 3$ we consider $p$-ordinary, Hilbert modular newforms $f$ of weight $k\\geq 2$ with associated $p$-adic Galois representations $\\rho_f$ and $\\mod{p^n}$ reductions $\\rho_{f,n}$. Under suitable hypotheses on the size of the image, we use deformation theory and modularity lifting to show that if the restrictions of $\\rho_{f,n} $ to decomposition groups above $p$ split then $f$ has a companion form $g$ modulo $p^n$ (in the sense that $\\rho_{f,n}\\sim \\rho_{g,n}\\otimes\\chi^{k-1}$)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.0181","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"}