{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:DTQSI6ICD2PXUMGPHGEOFFK2N3","short_pith_number":"pith:DTQSI6IC","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"},"canonical_sha256":"1ce12479021e9f7a30cf3988e2955a6eff99be0ae58cbe1dceec678ccbbff21c","source":{"kind":"arxiv","id":"1505.01216","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.01216","created_at":"2026-05-18T00:54:14Z"},{"alias_kind":"arxiv_version","alias_value":"1505.01216v2","created_at":"2026-05-18T00:54:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.01216","created_at":"2026-05-18T00:54:14Z"},{"alias_kind":"pith_short_12","alias_value":"DTQSI6ICD2PX","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_16","alias_value":"DTQSI6ICD2PXUMGP","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_8","alias_value":"DTQSI6IC","created_at":"2026-05-18T12:29:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:DTQSI6ICD2PXUMGPHGEOFFK2N3","target":"record","payload":{"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"},"canonical_sha256":"1ce12479021e9f7a30cf3988e2955a6eff99be0ae58cbe1dceec678ccbbff21c","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"},"source_kind":"arxiv","source_id":"1505.01216","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:54:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eylvjUR8OZ7vLMQVhhoLot4zDN5dpIB7EiE/zlV30l7S6JJ2l1QCij8yFee4hXFk2knKwBQKyLo4GupQuHI4Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T15:35:48.616645Z"},"content_sha256":"3b82d0470013bdb7cb5881293f89c91a672665799f638d233b98a72667838e87","schema_version":"1.0","event_id":"sha256:3b82d0470013bdb7cb5881293f89c91a672665799f638d233b98a72667838e87"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:DTQSI6ICD2PXUMGPHGEOFFK2N3","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:54:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CV8YvNnuEj11S1alkE8sjKDwsTenv62Je8Jb8y5NlwxL+J95PsWkeKvIcAExANLKf+zj4o0Kq4cDJMOWzWvIBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T15:35:48.616989Z"},"content_sha256":"c902038c7917bc0367b227af15c79e47513998c38c0f0b32759f5a3fe5249df5","schema_version":"1.0","event_id":"sha256:c902038c7917bc0367b227af15c79e47513998c38c0f0b32759f5a3fe5249df5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DTQSI6ICD2PXUMGPHGEOFFK2N3/bundle.json","state_url":"https://pith.science/pith/DTQSI6ICD2PXUMGPHGEOFFK2N3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DTQSI6ICD2PXUMGPHGEOFFK2N3/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-27T15:35:48Z","links":{"resolver":"https://pith.science/pith/DTQSI6ICD2PXUMGPHGEOFFK2N3","bundle":"https://pith.science/pith/DTQSI6ICD2PXUMGPHGEOFFK2N3/bundle.json","state":"https://pith.science/pith/DTQSI6ICD2PXUMGPHGEOFFK2N3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DTQSI6ICD2PXUMGPHGEOFFK2N3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:DTQSI6ICD2PXUMGPHGEOFFK2N3","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"4c4bc9a33f1abfd3163b73e0981bdf80955fd1e4240de7ed761de796a7b462f3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-05-05T23:08:14Z","title_canon_sha256":"a09cd549647fb3ccf549679b93f39dbbcc136fa78dea9b2e2d4fa4c412abe810"},"schema_version":"1.0","source":{"id":"1505.01216","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.01216","created_at":"2026-05-18T00:54:14Z"},{"alias_kind":"arxiv_version","alias_value":"1505.01216v2","created_at":"2026-05-18T00:54:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.01216","created_at":"2026-05-18T00:54:14Z"},{"alias_kind":"pith_short_12","alias_value":"DTQSI6ICD2PX","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_16","alias_value":"DTQSI6ICD2PXUMGP","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_8","alias_value":"DTQSI6IC","created_at":"2026-05-18T12:29:17Z"}],"graph_snapshots":[{"event_id":"sha256:c902038c7917bc0367b227af15c79e47513998c38c0f0b32759f5a3fe5249df5","target":"graph","created_at":"2026-05-18T00:54:14Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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 ","authors_text":"Jo\\\"el Bella\\\"iche","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-05-05T23:08:14Z","title":"Images of Pseudo-Representations and Coefficients of Modular Forms modulo p"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.01216","kind":"arxiv","version":2},"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":"sha256:3b82d0470013bdb7cb5881293f89c91a672665799f638d233b98a72667838e87","target":"record","created_at":"2026-05-18T00:54:14Z","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":"4c4bc9a33f1abfd3163b73e0981bdf80955fd1e4240de7ed761de796a7b462f3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-05-05T23:08:14Z","title_canon_sha256":"a09cd549647fb3ccf549679b93f39dbbcc136fa78dea9b2e2d4fa4c412abe810"},"schema_version":"1.0","source":{"id":"1505.01216","kind":"arxiv","version":2}},"canonical_sha256":"1ce12479021e9f7a30cf3988e2955a6eff99be0ae58cbe1dceec678ccbbff21c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1ce12479021e9f7a30cf3988e2955a6eff99be0ae58cbe1dceec678ccbbff21c","first_computed_at":"2026-05-18T00:54:14.036391Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:54:14.036391Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mFnAW3EQjN11p6DhFdDTlIEU3Wtgvwc/JDnPD4HbJAR9UG06UPD+iGO4PLn3aC9mF2fgWTMWYrOAM3vqaN24Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:54:14.036783Z","signed_message":"canonical_sha256_bytes"},"source_id":"1505.01216","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3b82d0470013bdb7cb5881293f89c91a672665799f638d233b98a72667838e87","sha256:c902038c7917bc0367b227af15c79e47513998c38c0f0b32759f5a3fe5249df5"],"state_sha256":"d133a62f0b935e552f9b5ae7bc1a68a3d0ad98bea2d39a06f0da3f12aaf1d99d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7fc+JDZxcp3HEEA6H45fx2ubIgkP/8V+YEH2YMhgBo6D2KHROp9l87ZPwQLRVRb9alm5m1DrGb3tHwcMJsLPCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T15:35:48.618828Z","bundle_sha256":"3f009b1e30811038f4da2f8a5417a4866e3eb56c7010ce02522e58cf66261475"}}