{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:CGSOIZGFRQUOXLMYHC5NOKPO2Y","short_pith_number":"pith:CGSOIZGF","canonical_record":{"source":{"id":"1410.5144","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-10-20T03:03:38Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"50064560000c0623e7f82b51f71035e7a8aaa37238fba26c6010b80505a5992e","abstract_canon_sha256":"7e3693a2b7680a0adf940a26cbe01fd885063105455e6d46fa93d2193992191a"},"schema_version":"1.0"},"canonical_sha256":"11a4e464c58c28ebad9838bad729eed6167bde4e1cfabf5a75626c14609c02d2","source":{"kind":"arxiv","id":"1410.5144","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.5144","created_at":"2026-05-17T23:50:39Z"},{"alias_kind":"arxiv_version","alias_value":"1410.5144v2","created_at":"2026-05-17T23:50:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.5144","created_at":"2026-05-17T23:50:39Z"},{"alias_kind":"pith_short_12","alias_value":"CGSOIZGFRQUO","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_16","alias_value":"CGSOIZGFRQUOXLMY","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_8","alias_value":"CGSOIZGF","created_at":"2026-05-18T12:28:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:CGSOIZGFRQUOXLMYHC5NOKPO2Y","target":"record","payload":{"canonical_record":{"source":{"id":"1410.5144","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-10-20T03:03:38Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"50064560000c0623e7f82b51f71035e7a8aaa37238fba26c6010b80505a5992e","abstract_canon_sha256":"7e3693a2b7680a0adf940a26cbe01fd885063105455e6d46fa93d2193992191a"},"schema_version":"1.0"},"canonical_sha256":"11a4e464c58c28ebad9838bad729eed6167bde4e1cfabf5a75626c14609c02d2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:50:39.068691Z","signature_b64":"xuzI4hH8wJ7WbUDGBIV8OL/n0y093lf+l53fmCiyOBxF8w68Q/1eo2/6RLN0cyK9e5k/PeNFfiR+epU5n8TYDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"11a4e464c58c28ebad9838bad729eed6167bde4e1cfabf5a75626c14609c02d2","last_reissued_at":"2026-05-17T23:50:39.068078Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:50:39.068078Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1410.5144","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:50:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c7pOn2Ohf8fpEjCelnwjGuFjRKGc4CQvuMTkoBORLn23lCvgWMN1V/0q/aAH1IM+/+7+uO4K2oXFCJQ50SDwBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T08:52:30.032142Z"},"content_sha256":"207fbbc99b7c78f9a423282347bd6ba1bc0fb2d1a253d395ecbcfde19de7bff3","schema_version":"1.0","event_id":"sha256:207fbbc99b7c78f9a423282347bd6ba1bc0fb2d1a253d395ecbcfde19de7bff3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:CGSOIZGFRQUOXLMYHC5NOKPO2Y","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Realization of groups with pairing as Jacobians of finite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.CO","authors_text":"David Jensen, Dhruv Ranganathan, Louis Gaudet, Nicholas Wawrykow, Theodore Weisman","submitted_at":"2014-10-20T03:03:38Z","abstract_excerpt":"We study which groups with pairing can occur as the Jacobian of a finite graph. We provide explicit constructions of graphs whose Jacobian realizes a large fraction of odd groups with a given pairing. Conditional on the generalized Riemann hypothesis, these constructions yield all groups with pairing of odd order, and unconditionally, they yield all groups with pairing whose prime factors are sufficiently large. For groups with pairing of even order, we provide a partial answer to this question, for a certain restricted class of pairings. Finally, we explore which finite abelian groups occur a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.5144","kind":"arxiv","version":2},"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":"2026-05-17T23:50:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2Xgjt3dfcitWZJYhYz309F/0tu4EiUUD/JIZ/q9SULDN5fmSaf163aooP+tWCR8UxjwwDNIVuzdvZhWB/5F3Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T08:52:30.032812Z"},"content_sha256":"ebd6beb23259a9eb43249f3c36aa3d171328d3f30568afafc2b562963567f385","schema_version":"1.0","event_id":"sha256:ebd6beb23259a9eb43249f3c36aa3d171328d3f30568afafc2b562963567f385"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CGSOIZGFRQUOXLMYHC5NOKPO2Y/bundle.json","state_url":"https://pith.science/pith/CGSOIZGFRQUOXLMYHC5NOKPO2Y/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CGSOIZGFRQUOXLMYHC5NOKPO2Y/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":"2026-05-30T08:52:30Z","links":{"resolver":"https://pith.science/pith/CGSOIZGFRQUOXLMYHC5NOKPO2Y","bundle":"https://pith.science/pith/CGSOIZGFRQUOXLMYHC5NOKPO2Y/bundle.json","state":"https://pith.science/pith/CGSOIZGFRQUOXLMYHC5NOKPO2Y/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CGSOIZGFRQUOXLMYHC5NOKPO2Y/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:CGSOIZGFRQUOXLMYHC5NOKPO2Y","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":"7e3693a2b7680a0adf940a26cbe01fd885063105455e6d46fa93d2193992191a","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-10-20T03:03:38Z","title_canon_sha256":"50064560000c0623e7f82b51f71035e7a8aaa37238fba26c6010b80505a5992e"},"schema_version":"1.0","source":{"id":"1410.5144","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.5144","created_at":"2026-05-17T23:50:39Z"},{"alias_kind":"arxiv_version","alias_value":"1410.5144v2","created_at":"2026-05-17T23:50:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.5144","created_at":"2026-05-17T23:50:39Z"},{"alias_kind":"pith_short_12","alias_value":"CGSOIZGFRQUO","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_16","alias_value":"CGSOIZGFRQUOXLMY","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_8","alias_value":"CGSOIZGF","created_at":"2026-05-18T12:28:22Z"}],"graph_snapshots":[{"event_id":"sha256:ebd6beb23259a9eb43249f3c36aa3d171328d3f30568afafc2b562963567f385","target":"graph","created_at":"2026-05-17T23:50:39Z","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":"We study which groups with pairing can occur as the Jacobian of a finite graph. We provide explicit constructions of graphs whose Jacobian realizes a large fraction of odd groups with a given pairing. Conditional on the generalized Riemann hypothesis, these constructions yield all groups with pairing of odd order, and unconditionally, they yield all groups with pairing whose prime factors are sufficiently large. For groups with pairing of even order, we provide a partial answer to this question, for a certain restricted class of pairings. Finally, we explore which finite abelian groups occur a","authors_text":"David Jensen, Dhruv Ranganathan, Louis Gaudet, Nicholas Wawrykow, Theodore Weisman","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-10-20T03:03:38Z","title":"Realization of groups with pairing as Jacobians of finite graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.5144","kind":"arxiv","version":2},"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":"sha256:207fbbc99b7c78f9a423282347bd6ba1bc0fb2d1a253d395ecbcfde19de7bff3","target":"record","created_at":"2026-05-17T23:50:39Z","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":"7e3693a2b7680a0adf940a26cbe01fd885063105455e6d46fa93d2193992191a","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-10-20T03:03:38Z","title_canon_sha256":"50064560000c0623e7f82b51f71035e7a8aaa37238fba26c6010b80505a5992e"},"schema_version":"1.0","source":{"id":"1410.5144","kind":"arxiv","version":2}},"canonical_sha256":"11a4e464c58c28ebad9838bad729eed6167bde4e1cfabf5a75626c14609c02d2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"11a4e464c58c28ebad9838bad729eed6167bde4e1cfabf5a75626c14609c02d2","first_computed_at":"2026-05-17T23:50:39.068078Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:50:39.068078Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xuzI4hH8wJ7WbUDGBIV8OL/n0y093lf+l53fmCiyOBxF8w68Q/1eo2/6RLN0cyK9e5k/PeNFfiR+epU5n8TYDA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:50:39.068691Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.5144","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:207fbbc99b7c78f9a423282347bd6ba1bc0fb2d1a253d395ecbcfde19de7bff3","sha256:ebd6beb23259a9eb43249f3c36aa3d171328d3f30568afafc2b562963567f385"],"state_sha256":"7b7a550b77d0ae8debab51f9ff7b36e80939bb808e821bdf7ad2bede2e9f8861"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TG9uzdlu6+w/Bhum4FtmWkjSCMK6X+6g/txDPm2DoHxGttqxZLpUahuKjasyiAZ2tDx3kEEsTUiSSvNRTwZMBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T08:52:30.036687Z","bundle_sha256":"b971472c9644f0c869bb0cd2634d886a6a1b69b0756bdca7ea8f9c1f506f9d85"}}