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If $k_f$ denotes the weight of a cusp form $f(z)= \\sum_{n\\geq 1} a_\\ell(f) e^{2\\pi i n z}$, then the inequality $$|a_\\ell(f) | \\leq 2 \\ell^{(k_f-1)/2},$$ predicted by the Ramanujan conjecture, is a strict inequality for almost all members $f$ of the family."},"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":"1801.03070","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-01-09T18:25:57Z","cross_cats_sorted":[],"title_canon_sha256":"5b2c5f299addde7468f9a541ec45b40d348f9c38c93b66830a210c3fbe9a6ffc","abstract_canon_sha256":"22269c2d2f43f5f4facaea177c29b829cd3f14e4d8a36465f03c8a3a9f5a5dd4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:26:14.089837Z","signature_b64":"5AMC4x2FW3kkolU4DscMeBDZpO7e+VPKQmDuMSMcvbLkgvStcy3pjejeM8U48LsfZAnLDkQqQsPYioicosXEAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"45379d9306a976ffec2b4ab82067ee090b6cfe5d0320ac76d892368edfa24861","last_reissued_at":"2026-05-18T00:26:14.089239Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:26:14.089239Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Frobenius semisimplicity in Hida families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jyoti Prakash Saha","submitted_at":"2018-01-09T18:25:57Z","abstract_excerpt":"Let $p\\geq 5$ be a prime and $\\ell\\neq p$ be a prime not dividing the tame level of a $p$-ordinary Hida family. We prove that the actions of the Frobenius element at $\\ell$ on the Galois representations attached to almost all arithmetic specializations are semisimple and non-scalar. If $k_f$ denotes the weight of a cusp form $f(z)= \\sum_{n\\geq 1} a_\\ell(f) e^{2\\pi i n z}$, then the inequality $$|a_\\ell(f) | \\leq 2 \\ell^{(k_f-1)/2},$$ predicted by the Ramanujan conjecture, is a strict inequality for almost all members $f$ of the family."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.03070","kind":"arxiv","version":2},"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":"1801.03070","created_at":"2026-05-18T00:26:14.089329+00:00"},{"alias_kind":"arxiv_version","alias_value":"1801.03070v2","created_at":"2026-05-18T00:26:14.089329+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.03070","created_at":"2026-05-18T00:26:14.089329+00:00"},{"alias_kind":"pith_short_12","alias_value":"IU3Z3EYGVF3P","created_at":"2026-05-18T12:32:31.084164+00:00"},{"alias_kind":"pith_short_16","alias_value":"IU3Z3EYGVF3P73BL","created_at":"2026-05-18T12:32:31.084164+00:00"},{"alias_kind":"pith_short_8","alias_value":"IU3Z3EYG","created_at":"2026-05-18T12:32:31.084164+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/IU3Z3EYGVF3P73BLJK4CAZ7OBE","json":"https://pith.science/pith/IU3Z3EYGVF3P73BLJK4CAZ7OBE.json","graph_json":"https://pith.science/api/pith-number/IU3Z3EYGVF3P73BLJK4CAZ7OBE/graph.json","events_json":"https://pith.science/api/pith-number/IU3Z3EYGVF3P73BLJK4CAZ7OBE/events.json","paper":"https://pith.science/paper/IU3Z3EYG"},"agent_actions":{"view_html":"https://pith.science/pith/IU3Z3EYGVF3P73BLJK4CAZ7OBE","download_json":"https://pith.science/pith/IU3Z3EYGVF3P73BLJK4CAZ7OBE.json","view_paper":"https://pith.science/paper/IU3Z3EYG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1801.03070&json=true","fetch_graph":"https://pith.science/api/pith-number/IU3Z3EYGVF3P73BLJK4CAZ7OBE/graph.json","fetch_events":"https://pith.science/api/pith-number/IU3Z3EYGVF3P73BLJK4CAZ7OBE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IU3Z3EYGVF3P73BLJK4CAZ7OBE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IU3Z3EYGVF3P73BLJK4CAZ7OBE/action/storage_attestation","attest_author":"https://pith.science/pith/IU3Z3EYGVF3P73BLJK4CAZ7OBE/action/author_attestation","sign_citation":"https://pith.science/pith/IU3Z3EYGVF3P73BLJK4CAZ7OBE/action/citation_signature","submit_replication":"https://pith.science/pith/IU3Z3EYGVF3P73BLJK4CAZ7OBE/action/replication_record"}},"created_at":"2026-05-18T00:26:14.089329+00:00","updated_at":"2026-05-18T00:26:14.089329+00:00"}