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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","authors_text":"Kevin Buzzard, Richard Taylor","cross_cats":[],"headline":"","license":"","primary_cat":"math.NT","submitted_at":"1999-05-01T00:00:00Z","title":"Companion forms and weight one forms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9905207","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:89065dc9119887ebaab3fc9332320096d379919b2bf92d97b82cbe30bb73e973","target":"record","created_at":"2026-05-18T01:05:33Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"28069753f16897828bdece47c79b7890c7624e71169cd403d59f4a8f6940eb12","cross_cats_sorted":[],"license":"","primary_cat":"math.NT","submitted_at":"1999-05-01T00:00:00Z","title_canon_sha256":"6b30e4d8e54564fe4f1c35d6c3f828d026a4e6c4e9603395a49f945ad2ad0307"},"schema_version":"1.0","source":{"id":"math/9905207","kind":"arxiv","version":1}},"canonical_sha256":"9e2715351ff642d1390836d590112e6c1ddd75097a7fe9f482e0be692799b2b3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9e2715351ff642d1390836d590112e6c1ddd75097a7fe9f482e0be692799b2b3","first_computed_at":"2026-05-18T01:05:33.015560Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:33.015560Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9wIXoWUtDzE1y4ZNnbX9U1gdOWjUJ4S52s+1ON+SfCdHIr5D+UBYMNTF0WpyrqWjad+/tOpOPiIfEOohF0TZAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:33.016226Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9905207","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:89065dc9119887ebaab3fc9332320096d379919b2bf92d97b82cbe30bb73e973","sha256:ef62d8fe19d8ff7ef91a7ee8c4fcfdaa63c2e9eb0eb5189c086f84672250493a"],"state_sha256":"ed498a948f4aa94a12e52930c9687da582895d6356bc3d36731c97ca7ac8494f"}