{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:5Q6VBGWBEXZ5QWQ3YLGXYYFTXE","short_pith_number":"pith:5Q6VBGWB","schema_version":"1.0","canonical_sha256":"ec3d509ac125f3d85a1bc2cd7c60b3b90ea113e3798ddd9ee08ad9595574ec21","source":{"kind":"arxiv","id":"1811.10359","version":1},"attestation_state":"computed","paper":{"title":"Modular cocycles and cup product","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Roelof Bruggeman, YoungJu Choie","submitted_at":"2018-11-26T13:29:31Z","abstract_excerpt":"We extend to positive real weights Haberland's formula giving a cohomological description of the Petersson scalar product of modular cusp forms of positive even weight. This relation is based on the cup product of an Eichler cocycle and a Knopp cocycle. We also consider the cup product of two Eichler cocycles attached to modular forms. In the classical context of integral weights at least $2$ this cup product is uninteresting. We show evidence that for real weights this cup product may very well be non-trivial. We approach the question whether the cup product is a non-trivial coinvariant by du"},"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":"1811.10359","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-11-26T13:29:31Z","cross_cats_sorted":[],"title_canon_sha256":"31dc08181c482469249c3fefd065016b64bcd595a71aabdd9f717bf51c4cf238","abstract_canon_sha256":"0c6a8b7b25cd911f9b342299f593b303584d9ed270e7222e76b6aa20ef66786d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:59:56.682857Z","signature_b64":"lHD+uotf40TsCRUhrKsWAtSXP8TreyB0BBsclalJ+vVVj0AVpFYagMZGfDV87QUmoEgmQNrdCvwrnyi4xcOGDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ec3d509ac125f3d85a1bc2cd7c60b3b90ea113e3798ddd9ee08ad9595574ec21","last_reissued_at":"2026-05-17T23:59:56.682358Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:59:56.682358Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Modular cocycles and cup product","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Roelof Bruggeman, YoungJu Choie","submitted_at":"2018-11-26T13:29:31Z","abstract_excerpt":"We extend to positive real weights Haberland's formula giving a cohomological description of the Petersson scalar product of modular cusp forms of positive even weight. This relation is based on the cup product of an Eichler cocycle and a Knopp cocycle. We also consider the cup product of two Eichler cocycles attached to modular forms. In the classical context of integral weights at least $2$ this cup product is uninteresting. We show evidence that for real weights this cup product may very well be non-trivial. We approach the question whether the cup product is a non-trivial coinvariant by du"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.10359","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1811.10359","created_at":"2026-05-17T23:59:56.682447+00:00"},{"alias_kind":"arxiv_version","alias_value":"1811.10359v1","created_at":"2026-05-17T23:59:56.682447+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.10359","created_at":"2026-05-17T23:59:56.682447+00:00"},{"alias_kind":"pith_short_12","alias_value":"5Q6VBGWBEXZ5","created_at":"2026-05-18T12:32:08.215937+00:00"},{"alias_kind":"pith_short_16","alias_value":"5Q6VBGWBEXZ5QWQ3","created_at":"2026-05-18T12:32:08.215937+00:00"},{"alias_kind":"pith_short_8","alias_value":"5Q6VBGWB","created_at":"2026-05-18T12:32:08.215937+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/5Q6VBGWBEXZ5QWQ3YLGXYYFTXE","json":"https://pith.science/pith/5Q6VBGWBEXZ5QWQ3YLGXYYFTXE.json","graph_json":"https://pith.science/api/pith-number/5Q6VBGWBEXZ5QWQ3YLGXYYFTXE/graph.json","events_json":"https://pith.science/api/pith-number/5Q6VBGWBEXZ5QWQ3YLGXYYFTXE/events.json","paper":"https://pith.science/paper/5Q6VBGWB"},"agent_actions":{"view_html":"https://pith.science/pith/5Q6VBGWBEXZ5QWQ3YLGXYYFTXE","download_json":"https://pith.science/pith/5Q6VBGWBEXZ5QWQ3YLGXYYFTXE.json","view_paper":"https://pith.science/paper/5Q6VBGWB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1811.10359&json=true","fetch_graph":"https://pith.science/api/pith-number/5Q6VBGWBEXZ5QWQ3YLGXYYFTXE/graph.json","fetch_events":"https://pith.science/api/pith-number/5Q6VBGWBEXZ5QWQ3YLGXYYFTXE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5Q6VBGWBEXZ5QWQ3YLGXYYFTXE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5Q6VBGWBEXZ5QWQ3YLGXYYFTXE/action/storage_attestation","attest_author":"https://pith.science/pith/5Q6VBGWBEXZ5QWQ3YLGXYYFTXE/action/author_attestation","sign_citation":"https://pith.science/pith/5Q6VBGWBEXZ5QWQ3YLGXYYFTXE/action/citation_signature","submit_replication":"https://pith.science/pith/5Q6VBGWBEXZ5QWQ3YLGXYYFTXE/action/replication_record"}},"created_at":"2026-05-17T23:59:56.682447+00:00","updated_at":"2026-05-17T23:59:56.682447+00:00"}