{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:LP74F7PGYMPGE5PC3WQVHJIWPS","short_pith_number":"pith:LP74F7PG","canonical_record":{"source":{"id":"1504.07913","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-04-29T16:09:08Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"31deab2df4f8dbc0b0650bcc4d8b9aa50cce6afe13d42c5e90595c3f2de7856f","abstract_canon_sha256":"8ec075062085ad668eee38bb8091dbeb4a38281488978e8d6ddfd17a91f4a90a"},"schema_version":"1.0"},"canonical_sha256":"5bffc2fde6c31e6275e2dda153a5167cbc45e6e2f6fd54ab0446e718a43391cf","source":{"kind":"arxiv","id":"1504.07913","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.07913","created_at":"2026-05-18T02:17:31Z"},{"alias_kind":"arxiv_version","alias_value":"1504.07913v1","created_at":"2026-05-18T02:17:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.07913","created_at":"2026-05-18T02:17:31Z"},{"alias_kind":"pith_short_12","alias_value":"LP74F7PGYMPG","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"LP74F7PGYMPGE5PC","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"LP74F7PG","created_at":"2026-05-18T12:29:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:LP74F7PGYMPGE5PC3WQVHJIWPS","target":"record","payload":{"canonical_record":{"source":{"id":"1504.07913","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-04-29T16:09:08Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"31deab2df4f8dbc0b0650bcc4d8b9aa50cce6afe13d42c5e90595c3f2de7856f","abstract_canon_sha256":"8ec075062085ad668eee38bb8091dbeb4a38281488978e8d6ddfd17a91f4a90a"},"schema_version":"1.0"},"canonical_sha256":"5bffc2fde6c31e6275e2dda153a5167cbc45e6e2f6fd54ab0446e718a43391cf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:17:31.918213Z","signature_b64":"VNiWyDYWu6GGulL7yAD6xGJ8q3z8tNiTlm7QmGol4rYZTylA07+wKE00y1ZnJ3w8tx6i8SiV8pRCrkh4mtsVCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5bffc2fde6c31e6275e2dda153a5167cbc45e6e2f6fd54ab0446e718a43391cf","last_reissued_at":"2026-05-18T02:17:31.917613Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:17:31.917613Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1504.07913","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:17:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dOaOMOpFEmSRic0yjq7uf1mIJ1jz066hQzURlNoyK50e3ziGKU+OunY0hNxF0nN0eNJxEjQgRcwrnrCMfLU9BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T10:42:59.604809Z"},"content_sha256":"80316e03c01996e343fb1b0b07a4603648e51ab5ba329b2fe287b09d91123873","schema_version":"1.0","event_id":"sha256:80316e03c01996e343fb1b0b07a4603648e51ab5ba329b2fe287b09d91123873"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:LP74F7PGYMPGE5PC3WQVHJIWPS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Picard groups of the stacks $Y_0(2)$ and $Y_0(3)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AG","authors_text":"Andrew Niles","submitted_at":"2015-04-29T16:09:08Z","abstract_excerpt":"We compute the Picard group of the stack of elliptic curves equipped with a cyclic subgroup of order two, and of the stack of elliptic curves equipped with a cyclic subgroup of order three, over any base scheme on which 6 is invertible. This generalizes a result of Fulton-Olsson, who computed the Picard group of the stack of elliptic curves (with no level structure) over a wide variety of base schemes."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.07913","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:17:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fyLtWZRBQxrGlS9APEIbl7uMAvDmPQ2DGfAmq8asPW4g/GAGNHBO0PRiiBsQvONrzeWA16fux5HGhORBdBuVBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T10:42:59.605261Z"},"content_sha256":"016928a27327d5b3762b85bacb035bc216ef930ac2a5c3dc2a4c4283c4d450cd","schema_version":"1.0","event_id":"sha256:016928a27327d5b3762b85bacb035bc216ef930ac2a5c3dc2a4c4283c4d450cd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LP74F7PGYMPGE5PC3WQVHJIWPS/bundle.json","state_url":"https://pith.science/pith/LP74F7PGYMPGE5PC3WQVHJIWPS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LP74F7PGYMPGE5PC3WQVHJIWPS/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":"2026-05-30T10:42:59Z","links":{"resolver":"https://pith.science/pith/LP74F7PGYMPGE5PC3WQVHJIWPS","bundle":"https://pith.science/pith/LP74F7PGYMPGE5PC3WQVHJIWPS/bundle.json","state":"https://pith.science/pith/LP74F7PGYMPGE5PC3WQVHJIWPS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LP74F7PGYMPGE5PC3WQVHJIWPS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:LP74F7PGYMPGE5PC3WQVHJIWPS","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":"8ec075062085ad668eee38bb8091dbeb4a38281488978e8d6ddfd17a91f4a90a","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-04-29T16:09:08Z","title_canon_sha256":"31deab2df4f8dbc0b0650bcc4d8b9aa50cce6afe13d42c5e90595c3f2de7856f"},"schema_version":"1.0","source":{"id":"1504.07913","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.07913","created_at":"2026-05-18T02:17:31Z"},{"alias_kind":"arxiv_version","alias_value":"1504.07913v1","created_at":"2026-05-18T02:17:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.07913","created_at":"2026-05-18T02:17:31Z"},{"alias_kind":"pith_short_12","alias_value":"LP74F7PGYMPG","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"LP74F7PGYMPGE5PC","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"LP74F7PG","created_at":"2026-05-18T12:29:29Z"}],"graph_snapshots":[{"event_id":"sha256:016928a27327d5b3762b85bacb035bc216ef930ac2a5c3dc2a4c4283c4d450cd","target":"graph","created_at":"2026-05-18T02:17:31Z","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 compute the Picard group of the stack of elliptic curves equipped with a cyclic subgroup of order two, and of the stack of elliptic curves equipped with a cyclic subgroup of order three, over any base scheme on which 6 is invertible. This generalizes a result of Fulton-Olsson, who computed the Picard group of the stack of elliptic curves (with no level structure) over a wide variety of base schemes.","authors_text":"Andrew Niles","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-04-29T16:09:08Z","title":"The Picard groups of the stacks $Y_0(2)$ and $Y_0(3)$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.07913","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":"sha256:80316e03c01996e343fb1b0b07a4603648e51ab5ba329b2fe287b09d91123873","target":"record","created_at":"2026-05-18T02:17:31Z","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":"8ec075062085ad668eee38bb8091dbeb4a38281488978e8d6ddfd17a91f4a90a","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-04-29T16:09:08Z","title_canon_sha256":"31deab2df4f8dbc0b0650bcc4d8b9aa50cce6afe13d42c5e90595c3f2de7856f"},"schema_version":"1.0","source":{"id":"1504.07913","kind":"arxiv","version":1}},"canonical_sha256":"5bffc2fde6c31e6275e2dda153a5167cbc45e6e2f6fd54ab0446e718a43391cf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5bffc2fde6c31e6275e2dda153a5167cbc45e6e2f6fd54ab0446e718a43391cf","first_computed_at":"2026-05-18T02:17:31.917613Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:17:31.917613Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VNiWyDYWu6GGulL7yAD6xGJ8q3z8tNiTlm7QmGol4rYZTylA07+wKE00y1ZnJ3w8tx6i8SiV8pRCrkh4mtsVCw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:17:31.918213Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.07913","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:80316e03c01996e343fb1b0b07a4603648e51ab5ba329b2fe287b09d91123873","sha256:016928a27327d5b3762b85bacb035bc216ef930ac2a5c3dc2a4c4283c4d450cd"],"state_sha256":"d6a17881a41683465a87015c28514bd836ac5dfc319282b9e52e0e07c484590a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wxJtGEvXNrRAFO9hajdDtnTYRkHQjQFco4O0UrA/U/CKEJRBU9f0gCrT675OZaFrbxvp8+Qpnv2AVgjyELZuAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T10:42:59.609714Z","bundle_sha256":"f559fee2aced910833570db4831934b2a768c43d41a0ef66617d04f1d1a1e8f8"}}