{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:TVYAOGJ3H75SRSYQ7RNASOIVKP","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":"9ce4e4e34096544465b93d034909c7db98fbe2607c0711aa09415fd9893d0feb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-15T07:40:49Z","title_canon_sha256":"b1cff98a42a7644bebb9d3285d5553c66a5ba39e168c6a55cacf84146b4d169f"},"schema_version":"1.0","source":{"id":"2607.13536","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.13536","created_at":"2026-07-16T01:22:52Z"},{"alias_kind":"arxiv_version","alias_value":"2607.13536v1","created_at":"2026-07-16T01:22:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.13536","created_at":"2026-07-16T01:22:52Z"},{"alias_kind":"pith_short_12","alias_value":"TVYAOGJ3H75S","created_at":"2026-07-16T01:22:52Z"},{"alias_kind":"pith_short_16","alias_value":"TVYAOGJ3H75SRSYQ","created_at":"2026-07-16T01:22:52Z"},{"alias_kind":"pith_short_8","alias_value":"TVYAOGJ3","created_at":"2026-07-16T01:22:52Z"}],"graph_snapshots":[{"event_id":"sha256:bbd5649bf97838e50dd3c8e7ee1827b2d95a90182b8a944b79163ec930c9d1e2","target":"graph","created_at":"2026-07-16T01:22:52Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2607.13536/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We give an explicit formula for the order of the rational cuspidal class group of the modular curve $X_1(N)$ for an arbitrary integer $N$. The proof relies on results of Streng on the group of modular units on $X_1(N)$, and requires computing a certain determinant involving the second Bernoulli polynomial. We also define a higher weight analogue of the cuspidal class group and speculate that its order is related to a similar determinant defined using a higher degree Bernoulli polynomial.","authors_text":"Fran\\c{c}ois Brunault (UMPA-ENSL)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-15T07:40:49Z","title":"Bernoulli determinants and cuspidal subgroups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.13536","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:ba5c2f6567eaed2044bdb3725d575ab929b4866915b575498f21d0e968f92edd","target":"record","created_at":"2026-07-16T01:22:52Z","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":"9ce4e4e34096544465b93d034909c7db98fbe2607c0711aa09415fd9893d0feb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-15T07:40:49Z","title_canon_sha256":"b1cff98a42a7644bebb9d3285d5553c66a5ba39e168c6a55cacf84146b4d169f"},"schema_version":"1.0","source":{"id":"2607.13536","kind":"arxiv","version":1}},"canonical_sha256":"9d7007193b3ffb28cb10fc5a09391553c9570563b4b09263c5b32acfc0912d35","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9d7007193b3ffb28cb10fc5a09391553c9570563b4b09263c5b32acfc0912d35","first_computed_at":"2026-07-16T01:22:52.297919Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-16T01:22:52.297919Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"krwHUdmP26mmJY2xK+0QqNDJ/3xPR0xWAc3zIVf3KFGg6AnKHmeFeCb19uO9nCpp8xTJ9+/j3OFglLZ69N17AQ==","signature_status":"signed_v1","signed_at":"2026-07-16T01:22:52.298851Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.13536","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ba5c2f6567eaed2044bdb3725d575ab929b4866915b575498f21d0e968f92edd","sha256:bbd5649bf97838e50dd3c8e7ee1827b2d95a90182b8a944b79163ec930c9d1e2"],"state_sha256":"7f3c078f78a946f9fa345b6ab48a905647a987d29709eb0229f6727a050899cb"}