{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:DTQSI6ICD2PXUMGPHGEOFFK2N3","short_pith_number":"pith:DTQSI6IC","schema_version":"1.0","canonical_sha256":"1ce12479021e9f7a30cf3988e2955a6eff99be0ae58cbe1dceec678ccbbff21c","source":{"kind":"arxiv","id":"1505.01216","version":2},"attestation_state":"computed","paper":{"title":"Images of Pseudo-Representations and Coefficients of Modular Forms modulo p","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jo\\\"el Bella\\\"iche","submitted_at":"2015-05-05T23:08:14Z","abstract_excerpt":"We describe the image of general families of two-dimensional representations over compact semi-local rings. Applying this description to the family carried by the universal Hecke algebra acting on the space of modular forms of level $N$ modulo a prime $p$, we prove new results about the coefficients of modular forms mod $p$. If $f=\\sum_{n=0}^\\infty a_n q^n$ is such a form, for which we can assume without loss of generality that $a_n=0$ if $(n,Np)>1$, calling $\\delta(f)$ the density of the set of primes $\\ell$ such that $a_\\ell \\neq 0$, we prove that $\\delta(f)>0$ provided that $f$ is not zero "},"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":"1505.01216","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-05-05T23:08:14Z","cross_cats_sorted":[],"title_canon_sha256":"a09cd549647fb3ccf549679b93f39dbbcc136fa78dea9b2e2d4fa4c412abe810","abstract_canon_sha256":"4c4bc9a33f1abfd3163b73e0981bdf80955fd1e4240de7ed761de796a7b462f3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:54:14.036783Z","signature_b64":"mFnAW3EQjN11p6DhFdDTlIEU3Wtgvwc/JDnPD4HbJAR9UG06UPD+iGO4PLn3aC9mF2fgWTMWYrOAM3vqaN24Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1ce12479021e9f7a30cf3988e2955a6eff99be0ae58cbe1dceec678ccbbff21c","last_reissued_at":"2026-05-18T00:54:14.036391Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:54:14.036391Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Images of Pseudo-Representations and Coefficients of Modular Forms modulo p","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jo\\\"el Bella\\\"iche","submitted_at":"2015-05-05T23:08:14Z","abstract_excerpt":"We describe the image of general families of two-dimensional representations over compact semi-local rings. Applying this description to the family carried by the universal Hecke algebra acting on the space of modular forms of level $N$ modulo a prime $p$, we prove new results about the coefficients of modular forms mod $p$. If $f=\\sum_{n=0}^\\infty a_n q^n$ is such a form, for which we can assume without loss of generality that $a_n=0$ if $(n,Np)>1$, calling $\\delta(f)$ the density of the set of primes $\\ell$ such that $a_\\ell \\neq 0$, we prove that $\\delta(f)>0$ provided that $f$ is not zero "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.01216","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":"1505.01216","created_at":"2026-05-18T00:54:14.036451+00:00"},{"alias_kind":"arxiv_version","alias_value":"1505.01216v2","created_at":"2026-05-18T00:54:14.036451+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.01216","created_at":"2026-05-18T00:54:14.036451+00:00"},{"alias_kind":"pith_short_12","alias_value":"DTQSI6ICD2PX","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_16","alias_value":"DTQSI6ICD2PXUMGP","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_8","alias_value":"DTQSI6IC","created_at":"2026-05-18T12:29:17.054201+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/DTQSI6ICD2PXUMGPHGEOFFK2N3","json":"https://pith.science/pith/DTQSI6ICD2PXUMGPHGEOFFK2N3.json","graph_json":"https://pith.science/api/pith-number/DTQSI6ICD2PXUMGPHGEOFFK2N3/graph.json","events_json":"https://pith.science/api/pith-number/DTQSI6ICD2PXUMGPHGEOFFK2N3/events.json","paper":"https://pith.science/paper/DTQSI6IC"},"agent_actions":{"view_html":"https://pith.science/pith/DTQSI6ICD2PXUMGPHGEOFFK2N3","download_json":"https://pith.science/pith/DTQSI6ICD2PXUMGPHGEOFFK2N3.json","view_paper":"https://pith.science/paper/DTQSI6IC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1505.01216&json=true","fetch_graph":"https://pith.science/api/pith-number/DTQSI6ICD2PXUMGPHGEOFFK2N3/graph.json","fetch_events":"https://pith.science/api/pith-number/DTQSI6ICD2PXUMGPHGEOFFK2N3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DTQSI6ICD2PXUMGPHGEOFFK2N3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DTQSI6ICD2PXUMGPHGEOFFK2N3/action/storage_attestation","attest_author":"https://pith.science/pith/DTQSI6ICD2PXUMGPHGEOFFK2N3/action/author_attestation","sign_citation":"https://pith.science/pith/DTQSI6ICD2PXUMGPHGEOFFK2N3/action/citation_signature","submit_replication":"https://pith.science/pith/DTQSI6ICD2PXUMGPHGEOFFK2N3/action/replication_record"}},"created_at":"2026-05-18T00:54:14.036451+00:00","updated_at":"2026-05-18T00:54:14.036451+00:00"}