{"paper":{"title":"Modularity of residual Galois extensions and the Eisenstein ideal","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Krzysztof Klosin, Tobias Berger","submitted_at":"2018-10-17T21:16:51Z","abstract_excerpt":"For a totally real field $F$, a finite extension $\\mathbf{F}$ of $\\mathbf{F}_p$ and a Galois character $\\chi: G_F \\to \\mathbf{F}^{\\times}$ unramified away from a finite set of places $\\Sigma \\supset \\{\\mathfrak{p} \\mid p\\}$ consider the Bloch-Kato Selmer group $H:=H^1_{\\Sigma}(F, \\chi^{-1})$. In an earlier paper of the authors it was proved that the number $d$ of isomorphism classes of (non-semisimple, reducible) residual representations $\\overline{\\rho}$ giving rise to lines in $H$ which are modular by some $\\rho_f$ (also unramified outside $\\Sigma$) satisfies $d \\geq n:= \\dim_{\\mathbf{F}} H$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.07808","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":""},"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"}