{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:WZKSE4J4XOKUF3ZEPYUZH22Z3X","short_pith_number":"pith:WZKSE4J4","canonical_record":{"source":{"id":"1704.06941","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-04-23T15:37:33Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"1922aa0bdb4d6bf2c830410f6bd7a9c239c1932d1712324f884184c51e077fd8","abstract_canon_sha256":"10189313a6afb2b5219c606dbe78f9489b0a399ad6511fced8f74487c6871dfe"},"schema_version":"1.0"},"canonical_sha256":"b65522713cbb9542ef247e2993eb59ddf56e2185d8de6c8007d960e0e6306123","source":{"kind":"arxiv","id":"1704.06941","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1704.06941","created_at":"2026-05-18T00:45:55Z"},{"alias_kind":"arxiv_version","alias_value":"1704.06941v1","created_at":"2026-05-18T00:45:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.06941","created_at":"2026-05-18T00:45:55Z"},{"alias_kind":"pith_short_12","alias_value":"WZKSE4J4XOKU","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_16","alias_value":"WZKSE4J4XOKUF3ZE","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_8","alias_value":"WZKSE4J4","created_at":"2026-05-18T12:31:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:WZKSE4J4XOKUF3ZEPYUZH22Z3X","target":"record","payload":{"canonical_record":{"source":{"id":"1704.06941","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-04-23T15:37:33Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"1922aa0bdb4d6bf2c830410f6bd7a9c239c1932d1712324f884184c51e077fd8","abstract_canon_sha256":"10189313a6afb2b5219c606dbe78f9489b0a399ad6511fced8f74487c6871dfe"},"schema_version":"1.0"},"canonical_sha256":"b65522713cbb9542ef247e2993eb59ddf56e2185d8de6c8007d960e0e6306123","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:45:55.262855Z","signature_b64":"H50QeUTzPYBaUqyVib6TIQfur2Q1RritdGFLmC0eycDm5Wn7gJ2fkzGcUw5hZz6x2+e2PeoqryCJ7EuRk5y4Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b65522713cbb9542ef247e2993eb59ddf56e2185d8de6c8007d960e0e6306123","last_reissued_at":"2026-05-18T00:45:55.262215Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:45:55.262215Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1704.06941","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":"2026-05-18T00:45:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AvqXKiqU9fvRCncwmdIPDjDD9YzdxZRrb9ooYMyUPr3RYnomX7te7TL3KHvBkdlNVMTY9fvAaWsKGl2ZWTBnDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T02:22:06.612950Z"},"content_sha256":"4217ea9b7bbc5f4ca2efd64ef021f4d771c1384aaf4df1584b7a4dbcb677a089","schema_version":"1.0","event_id":"sha256:4217ea9b7bbc5f4ca2efd64ef021f4d771c1384aaf4df1584b7a4dbcb677a089"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:WZKSE4J4XOKUF3ZEPYUZH22Z3X","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Congruence formulae for Legendre modular polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Adel Betina, Emmanuel Lecouturier","submitted_at":"2017-04-23T15:37:33Z","abstract_excerpt":"Let $p\\geq 5$ be a prime number. We generalize the results of E. de Shalit about supersingular $j$-invariants in characteristic $p$.\n  We consider supersingular elliptic curves with a basis of $2$-torsion over $\\overline{\\mathbf{F}}_p$, or equivalently supersingular Legendre $\\lambda$-invariants. Let $F_p(X,Y) \\in \\mathbf{Z}[X,Y]$ be the $p$-th modular polynomial for $\\lambda$-invariants. A simple generalization of Kronecker's classical congruence shows that $R(X):=\\frac{F_p(X,X^{p})}{p}$ is in $\\mathbf{Z}[X]$. We give a formula for $R(\\lambda)$ if $\\lambda$ is a supersingular. This formula is"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.06941","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":"2026-05-18T00:45:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O9+S6W4eOSlg4baEaY6/InucgHpI3W/aF6WPZLKDDelopq249gK93qOPLcOPA2Gv/IElDotax3AT8t/3scNzAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T02:22:06.613284Z"},"content_sha256":"b6d49738aef216ad0ef5c64032b7b0c78ce0feb905ff1988121cde26a62b08ba","schema_version":"1.0","event_id":"sha256:b6d49738aef216ad0ef5c64032b7b0c78ce0feb905ff1988121cde26a62b08ba"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WZKSE4J4XOKUF3ZEPYUZH22Z3X/bundle.json","state_url":"https://pith.science/pith/WZKSE4J4XOKUF3ZEPYUZH22Z3X/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WZKSE4J4XOKUF3ZEPYUZH22Z3X/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":"2026-06-29T02:22:06Z","links":{"resolver":"https://pith.science/pith/WZKSE4J4XOKUF3ZEPYUZH22Z3X","bundle":"https://pith.science/pith/WZKSE4J4XOKUF3ZEPYUZH22Z3X/bundle.json","state":"https://pith.science/pith/WZKSE4J4XOKUF3ZEPYUZH22Z3X/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WZKSE4J4XOKUF3ZEPYUZH22Z3X/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:WZKSE4J4XOKUF3ZEPYUZH22Z3X","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":"10189313a6afb2b5219c606dbe78f9489b0a399ad6511fced8f74487c6871dfe","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-04-23T15:37:33Z","title_canon_sha256":"1922aa0bdb4d6bf2c830410f6bd7a9c239c1932d1712324f884184c51e077fd8"},"schema_version":"1.0","source":{"id":"1704.06941","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1704.06941","created_at":"2026-05-18T00:45:55Z"},{"alias_kind":"arxiv_version","alias_value":"1704.06941v1","created_at":"2026-05-18T00:45:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.06941","created_at":"2026-05-18T00:45:55Z"},{"alias_kind":"pith_short_12","alias_value":"WZKSE4J4XOKU","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_16","alias_value":"WZKSE4J4XOKUF3ZE","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_8","alias_value":"WZKSE4J4","created_at":"2026-05-18T12:31:53Z"}],"graph_snapshots":[{"event_id":"sha256:b6d49738aef216ad0ef5c64032b7b0c78ce0feb905ff1988121cde26a62b08ba","target":"graph","created_at":"2026-05-18T00:45:55Z","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":"Let $p\\geq 5$ be a prime number. We generalize the results of E. de Shalit about supersingular $j$-invariants in characteristic $p$.\n  We consider supersingular elliptic curves with a basis of $2$-torsion over $\\overline{\\mathbf{F}}_p$, or equivalently supersingular Legendre $\\lambda$-invariants. Let $F_p(X,Y) \\in \\mathbf{Z}[X,Y]$ be the $p$-th modular polynomial for $\\lambda$-invariants. A simple generalization of Kronecker's classical congruence shows that $R(X):=\\frac{F_p(X,X^{p})}{p}$ is in $\\mathbf{Z}[X]$. We give a formula for $R(\\lambda)$ if $\\lambda$ is a supersingular. This formula is","authors_text":"Adel Betina, Emmanuel Lecouturier","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-04-23T15:37:33Z","title":"Congruence formulae for Legendre modular polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.06941","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":"sha256:4217ea9b7bbc5f4ca2efd64ef021f4d771c1384aaf4df1584b7a4dbcb677a089","target":"record","created_at":"2026-05-18T00:45:55Z","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":"10189313a6afb2b5219c606dbe78f9489b0a399ad6511fced8f74487c6871dfe","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-04-23T15:37:33Z","title_canon_sha256":"1922aa0bdb4d6bf2c830410f6bd7a9c239c1932d1712324f884184c51e077fd8"},"schema_version":"1.0","source":{"id":"1704.06941","kind":"arxiv","version":1}},"canonical_sha256":"b65522713cbb9542ef247e2993eb59ddf56e2185d8de6c8007d960e0e6306123","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b65522713cbb9542ef247e2993eb59ddf56e2185d8de6c8007d960e0e6306123","first_computed_at":"2026-05-18T00:45:55.262215Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:45:55.262215Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"H50QeUTzPYBaUqyVib6TIQfur2Q1RritdGFLmC0eycDm5Wn7gJ2fkzGcUw5hZz6x2+e2PeoqryCJ7EuRk5y4Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T00:45:55.262855Z","signed_message":"canonical_sha256_bytes"},"source_id":"1704.06941","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4217ea9b7bbc5f4ca2efd64ef021f4d771c1384aaf4df1584b7a4dbcb677a089","sha256:b6d49738aef216ad0ef5c64032b7b0c78ce0feb905ff1988121cde26a62b08ba"],"state_sha256":"01c73dcf10708d3b5383eb9d582454422ce5765182c093a2bdd3aef4bb723215"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iJMFv2d3MhiF1TT8TVpMYdMtDf7rj609AG2jidellLhOg0m8d7vk7+emUJBmWPD2C0qskFlkAvc1cEI3GxaYDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T02:22:06.615124Z","bundle_sha256":"afd85a2f3ccc68bc77d869ef4383086a4724ab5f1ec1d91dd7ffedfa52984bb9"}}