{"paper":{"title":"Companion forms and weight one forms","license":"","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Kevin Buzzard, Richard Taylor","submitted_at":"1999-05-01T00:00:00Z","abstract_excerpt":"In this paper we prove the following theorem. Let L/\\Q_p be a finite extension with ring of integers O_L and maximal ideal lambda.\n  Theorem 1. Suppose that p >= 5. Suppose also that \\rho:G_\\Q -> GL_2(O_L) is a continuous representation satisfying the following conditions.\n  1. \\rho ramifies at only finitely many primes.\n  2. \\rho mod \\lambda is modular and absolutely irreducible.\n  3. \\rho is unramified at p and \\rho(Frob_p) has eigenvalues \\alpha and \\beta with distinct reductions modulo \\lambda.\n  Then there exists a classical weight one eigenform\n  f = \\sum_{n=1}^\\infty a_m(f) q^m\n  and an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9905207","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"}