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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math/9905207","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"1999-05-01T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"6b30e4d8e54564fe4f1c35d6c3f828d026a4e6c4e9603395a49f945ad2ad0307","abstract_canon_sha256":"28069753f16897828bdece47c79b7890c7624e71169cd403d59f4a8f6940eb12"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:33.016226Z","signature_b64":"9wIXoWUtDzE1y4ZNnbX9U1gdOWjUJ4S52s+1ON+SfCdHIr5D+UBYMNTF0WpyrqWjad+/tOpOPiIfEOohF0TZAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9e2715351ff642d1390836d590112e6c1ddd75097a7fe9f482e0be692799b2b3","last_reissued_at":"2026-05-18T01:05:33.015560Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:33.015560Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/9905207","created_at":"2026-05-18T01:05:33.015664+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/9905207v1","created_at":"2026-05-18T01:05:33.015664+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9905207","created_at":"2026-05-18T01:05:33.015664+00:00"},{"alias_kind":"pith_short_12","alias_value":"TYTRKNI76ZBN","created_at":"2026-05-18T12:25:49.631198+00:00"},{"alias_kind":"pith_short_16","alias_value":"TYTRKNI76ZBNCOII","created_at":"2026-05-18T12:25:49.631198+00:00"},{"alias_kind":"pith_short_8","alias_value":"TYTRKNI7","created_at":"2026-05-18T12:25:49.631198+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TYTRKNI76ZBNCOIIG3KZAEJONQ","json":"https://pith.science/pith/TYTRKNI76ZBNCOIIG3KZAEJONQ.json","graph_json":"https://pith.science/api/pith-number/TYTRKNI76ZBNCOIIG3KZAEJONQ/graph.json","events_json":"https://pith.science/api/pith-number/TYTRKNI76ZBNCOIIG3KZAEJONQ/events.json","paper":"https://pith.science/paper/TYTRKNI7"},"agent_actions":{"view_html":"https://pith.science/pith/TYTRKNI76ZBNCOIIG3KZAEJONQ","download_json":"https://pith.science/pith/TYTRKNI76ZBNCOIIG3KZAEJONQ.json","view_paper":"https://pith.science/paper/TYTRKNI7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/9905207&json=true","fetch_graph":"https://pith.science/api/pith-number/TYTRKNI76ZBNCOIIG3KZAEJONQ/graph.json","fetch_events":"https://pith.science/api/pith-number/TYTRKNI76ZBNCOIIG3KZAEJONQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TYTRKNI76ZBNCOIIG3KZAEJONQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TYTRKNI76ZBNCOIIG3KZAEJONQ/action/storage_attestation","attest_author":"https://pith.science/pith/TYTRKNI76ZBNCOIIG3KZAEJONQ/action/author_attestation","sign_citation":"https://pith.science/pith/TYTRKNI76ZBNCOIIG3KZAEJONQ/action/citation_signature","submit_replication":"https://pith.science/pith/TYTRKNI76ZBNCOIIG3KZAEJONQ/action/replication_record"}},"created_at":"2026-05-18T01:05:33.015664+00:00","updated_at":"2026-05-18T01:05:33.015664+00:00"}