{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:5LMW2WI2UGRJAE53QW7H526NJX","short_pith_number":"pith:5LMW2WI2","canonical_record":{"source":{"id":"1305.4505","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-05-20T11:53:38Z","cross_cats_sorted":[],"title_canon_sha256":"9b8747322c0fb2a06dcad0d7b8f78e2f3b492e20cee0e6eb79b71e07e1ff24e3","abstract_canon_sha256":"e5c8f98441ad34f1f3fe1465901fa8d83bf5484a592ffc0d0aa9e81b1618fc3a"},"schema_version":"1.0"},"canonical_sha256":"ead96d591aa1a29013bb85be7eebcd4de2cb38f69a5eb0e1d268765af962ae15","source":{"kind":"arxiv","id":"1305.4505","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.4505","created_at":"2026-05-18T03:25:23Z"},{"alias_kind":"arxiv_version","alias_value":"1305.4505v1","created_at":"2026-05-18T03:25:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.4505","created_at":"2026-05-18T03:25:23Z"},{"alias_kind":"pith_short_12","alias_value":"5LMW2WI2UGRJ","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_16","alias_value":"5LMW2WI2UGRJAE53","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_8","alias_value":"5LMW2WI2","created_at":"2026-05-18T12:27:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:5LMW2WI2UGRJAE53QW7H526NJX","target":"record","payload":{"canonical_record":{"source":{"id":"1305.4505","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-05-20T11:53:38Z","cross_cats_sorted":[],"title_canon_sha256":"9b8747322c0fb2a06dcad0d7b8f78e2f3b492e20cee0e6eb79b71e07e1ff24e3","abstract_canon_sha256":"e5c8f98441ad34f1f3fe1465901fa8d83bf5484a592ffc0d0aa9e81b1618fc3a"},"schema_version":"1.0"},"canonical_sha256":"ead96d591aa1a29013bb85be7eebcd4de2cb38f69a5eb0e1d268765af962ae15","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:25:23.535928Z","signature_b64":"am3QnR0IUQLPkYb7u7irOjnQnsfuV9wU7U3faoYeLIbfFCTwbE6qlaLnMZsqF/NdMP0U/nGyN7aNrL3bavTRAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ead96d591aa1a29013bb85be7eebcd4de2cb38f69a5eb0e1d268765af962ae15","last_reissued_at":"2026-05-18T03:25:23.535434Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:25:23.535434Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1305.4505","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":"2026-05-18T03:25:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Hk1p4GNWVlr0PdILD8eEjKtDOF+ibW5imQlFgHI0p+VQ8FI/iuQJd6U519Fg6rzcAbQzLuNoQUVjdHhh6zqXCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T00:15:47.578020Z"},"content_sha256":"4f6c8cc588a281dff380a596a06f309481e366a4d39a8cb00afc42fd33b0a6fe","schema_version":"1.0","event_id":"sha256:4f6c8cc588a281dff380a596a06f309481e366a4d39a8cb00afc42fd33b0a6fe"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:5LMW2WI2UGRJAE53QW7H526NJX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Computing points on modular curves over finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jinxiang Zeng","submitted_at":"2013-05-20T11:53:38Z","abstract_excerpt":"In this paper, we present a probabilistic algorithm to compute the number of $\\mathbb{F}_p$-points of modular curve $X_1(n)$. Under the Generalized Riemann Hypothesis(GRH), the algorithm takes $\\textrm{O}(n^{56+\\delta+\\epsilon}\\log^{9+\\epsilon} p)$ bit operations, where $\\delta$ is an absolute constant and $\\epsilon$ is any positive real number. As an application, we can compute $#X_1(17)(\\mathbb{F}_p)\\textrm{mod} 17$ for huge primes $p$. For example, we have $#X_1(17)(\\mathbb{F}_{10^{1000}+1357})\\textrm{mod} 17=3$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.4505","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":"2026-05-18T03:25:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XEGtAiYSig+71lcAdtuQGlOZ88Gu8+xmwY2PY31GGogn6Dx/GQo4hffYsiLgM+rLDz1op3UXtBLgZTLHMWBJCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T00:15:47.578683Z"},"content_sha256":"922a520dbf95295d4451e99d362d50b80b00b8c42174b324d18da06e97be1851","schema_version":"1.0","event_id":"sha256:922a520dbf95295d4451e99d362d50b80b00b8c42174b324d18da06e97be1851"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5LMW2WI2UGRJAE53QW7H526NJX/bundle.json","state_url":"https://pith.science/pith/5LMW2WI2UGRJAE53QW7H526NJX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5LMW2WI2UGRJAE53QW7H526NJX/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":"2026-06-10T00:15:47Z","links":{"resolver":"https://pith.science/pith/5LMW2WI2UGRJAE53QW7H526NJX","bundle":"https://pith.science/pith/5LMW2WI2UGRJAE53QW7H526NJX/bundle.json","state":"https://pith.science/pith/5LMW2WI2UGRJAE53QW7H526NJX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5LMW2WI2UGRJAE53QW7H526NJX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:5LMW2WI2UGRJAE53QW7H526NJX","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":"e5c8f98441ad34f1f3fe1465901fa8d83bf5484a592ffc0d0aa9e81b1618fc3a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-05-20T11:53:38Z","title_canon_sha256":"9b8747322c0fb2a06dcad0d7b8f78e2f3b492e20cee0e6eb79b71e07e1ff24e3"},"schema_version":"1.0","source":{"id":"1305.4505","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.4505","created_at":"2026-05-18T03:25:23Z"},{"alias_kind":"arxiv_version","alias_value":"1305.4505v1","created_at":"2026-05-18T03:25:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.4505","created_at":"2026-05-18T03:25:23Z"},{"alias_kind":"pith_short_12","alias_value":"5LMW2WI2UGRJ","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_16","alias_value":"5LMW2WI2UGRJAE53","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_8","alias_value":"5LMW2WI2","created_at":"2026-05-18T12:27:34Z"}],"graph_snapshots":[{"event_id":"sha256:922a520dbf95295d4451e99d362d50b80b00b8c42174b324d18da06e97be1851","target":"graph","created_at":"2026-05-18T03:25:23Z","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":"In this paper, we present a probabilistic algorithm to compute the number of $\\mathbb{F}_p$-points of modular curve $X_1(n)$. Under the Generalized Riemann Hypothesis(GRH), the algorithm takes $\\textrm{O}(n^{56+\\delta+\\epsilon}\\log^{9+\\epsilon} p)$ bit operations, where $\\delta$ is an absolute constant and $\\epsilon$ is any positive real number. As an application, we can compute $#X_1(17)(\\mathbb{F}_p)\\textrm{mod} 17$ for huge primes $p$. For example, we have $#X_1(17)(\\mathbb{F}_{10^{1000}+1357})\\textrm{mod} 17=3$.","authors_text":"Jinxiang Zeng","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-05-20T11:53:38Z","title":"Computing points on modular curves over finite fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.4505","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":"sha256:4f6c8cc588a281dff380a596a06f309481e366a4d39a8cb00afc42fd33b0a6fe","target":"record","created_at":"2026-05-18T03:25:23Z","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":"e5c8f98441ad34f1f3fe1465901fa8d83bf5484a592ffc0d0aa9e81b1618fc3a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-05-20T11:53:38Z","title_canon_sha256":"9b8747322c0fb2a06dcad0d7b8f78e2f3b492e20cee0e6eb79b71e07e1ff24e3"},"schema_version":"1.0","source":{"id":"1305.4505","kind":"arxiv","version":1}},"canonical_sha256":"ead96d591aa1a29013bb85be7eebcd4de2cb38f69a5eb0e1d268765af962ae15","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ead96d591aa1a29013bb85be7eebcd4de2cb38f69a5eb0e1d268765af962ae15","first_computed_at":"2026-05-18T03:25:23.535434Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:25:23.535434Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"am3QnR0IUQLPkYb7u7irOjnQnsfuV9wU7U3faoYeLIbfFCTwbE6qlaLnMZsqF/NdMP0U/nGyN7aNrL3bavTRAA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:25:23.535928Z","signed_message":"canonical_sha256_bytes"},"source_id":"1305.4505","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4f6c8cc588a281dff380a596a06f309481e366a4d39a8cb00afc42fd33b0a6fe","sha256:922a520dbf95295d4451e99d362d50b80b00b8c42174b324d18da06e97be1851"],"state_sha256":"65204c2156d67f87610efac4c65ebc28f60dc14a2975308467c8cfa91ab56cc8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NQjnhQFeMzx6yqWNBddXBruy7yjwYGt7/mVky/zIMSB9FsO4dxdbdaQt7LhugK4t25IdXILZND/oh31qe1eLAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T00:15:47.582643Z","bundle_sha256":"e0b28556c8683b80edc55b1fb356e68e656d798b2554f3d28d8e27edda3f8110"}}