{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:OJ7NH2FZGRBXKFD45FGSUEJKLJ","short_pith_number":"pith:OJ7NH2FZ","schema_version":"1.0","canonical_sha256":"727ed3e8b9344375147ce94d2a112a5a6dd5d55e30e59814d8afec63969cd950","source":{"kind":"arxiv","id":"1511.05278","version":1},"attestation_state":"computed","paper":{"title":"Icosahedron, exceptional singularities and modular forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.RT"],"primary_cat":"math.NT","authors_text":"Lei Yang","submitted_at":"2015-11-17T06:19:38Z","abstract_excerpt":"We find that the equation of $E_8$-singularity possesses two distinct symmetry groups and modular parametrizations. One is the classical icosahedral equation with icosahedral symmetry, the associated modular forms are theta constants of order five. The other is given by the group $\\text{PSL}(2, 13)$, the associated modular forms are theta constants of order $13$. As a consequence, we show that $E_8$ is not uniquely determined by the icosahedron. This solves a problem of Brieskorn in his ICM 1970 talk on the mysterious relation between exotic spheres, the icosahedron and $E_8$. Simultaneously, "},"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":"1511.05278","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-11-17T06:19:38Z","cross_cats_sorted":["math.AG","math.RT"],"title_canon_sha256":"74c61f59071ca1aadef607fb7d3e919ea09e040a569c87091ace7a8af6f38a29","abstract_canon_sha256":"f4538e9e41262aa05e4f6af44cdbed981c7b0d731c7450a41a197413c9a78272"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:26:49.598012Z","signature_b64":"6d+F1+ExKVbZbFueIBrPesAgfDZ2P5RnC5GtvMFy3soLfbi5q2WttnAsjR4396rj0UUWIeXrZ8s7ovsgeGRkBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"727ed3e8b9344375147ce94d2a112a5a6dd5d55e30e59814d8afec63969cd950","last_reissued_at":"2026-05-18T01:26:49.597416Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:26:49.597416Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Icosahedron, exceptional singularities and modular forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.RT"],"primary_cat":"math.NT","authors_text":"Lei Yang","submitted_at":"2015-11-17T06:19:38Z","abstract_excerpt":"We find that the equation of $E_8$-singularity possesses two distinct symmetry groups and modular parametrizations. One is the classical icosahedral equation with icosahedral symmetry, the associated modular forms are theta constants of order five. The other is given by the group $\\text{PSL}(2, 13)$, the associated modular forms are theta constants of order $13$. As a consequence, we show that $E_8$ is not uniquely determined by the icosahedron. This solves a problem of Brieskorn in his ICM 1970 talk on the mysterious relation between exotic spheres, the icosahedron and $E_8$. Simultaneously, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.05278","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":"1511.05278","created_at":"2026-05-18T01:26:49.597525+00:00"},{"alias_kind":"arxiv_version","alias_value":"1511.05278v1","created_at":"2026-05-18T01:26:49.597525+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.05278","created_at":"2026-05-18T01:26:49.597525+00:00"},{"alias_kind":"pith_short_12","alias_value":"OJ7NH2FZGRBX","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_16","alias_value":"OJ7NH2FZGRBXKFD4","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_8","alias_value":"OJ7NH2FZ","created_at":"2026-05-18T12:29:34.919912+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/OJ7NH2FZGRBXKFD45FGSUEJKLJ","json":"https://pith.science/pith/OJ7NH2FZGRBXKFD45FGSUEJKLJ.json","graph_json":"https://pith.science/api/pith-number/OJ7NH2FZGRBXKFD45FGSUEJKLJ/graph.json","events_json":"https://pith.science/api/pith-number/OJ7NH2FZGRBXKFD45FGSUEJKLJ/events.json","paper":"https://pith.science/paper/OJ7NH2FZ"},"agent_actions":{"view_html":"https://pith.science/pith/OJ7NH2FZGRBXKFD45FGSUEJKLJ","download_json":"https://pith.science/pith/OJ7NH2FZGRBXKFD45FGSUEJKLJ.json","view_paper":"https://pith.science/paper/OJ7NH2FZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1511.05278&json=true","fetch_graph":"https://pith.science/api/pith-number/OJ7NH2FZGRBXKFD45FGSUEJKLJ/graph.json","fetch_events":"https://pith.science/api/pith-number/OJ7NH2FZGRBXKFD45FGSUEJKLJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OJ7NH2FZGRBXKFD45FGSUEJKLJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OJ7NH2FZGRBXKFD45FGSUEJKLJ/action/storage_attestation","attest_author":"https://pith.science/pith/OJ7NH2FZGRBXKFD45FGSUEJKLJ/action/author_attestation","sign_citation":"https://pith.science/pith/OJ7NH2FZGRBXKFD45FGSUEJKLJ/action/citation_signature","submit_replication":"https://pith.science/pith/OJ7NH2FZGRBXKFD45FGSUEJKLJ/action/replication_record"}},"created_at":"2026-05-18T01:26:49.597525+00:00","updated_at":"2026-05-18T01:26:49.597525+00:00"}