{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:ZIHKZY55756RKZHA4URAL2HDWR","short_pith_number":"pith:ZIHKZY55","canonical_record":{"source":{"id":"1912.09860","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-12-20T14:59:30Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"b4d47fef204efdf600d2eb95c9129eb45091a9e2f55389856b796a19f79599e4","abstract_canon_sha256":"093ae02c459afa46a37eb978f968524028aa705072ebb89ef637f78c55b19724"},"schema_version":"1.0"},"canonical_sha256":"ca0eace3bdff7d1564e0e52205e8e3b44899166e6f712e063fa0f850ebb30364","source":{"kind":"arxiv","id":"1912.09860","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1912.09860","created_at":"2026-07-05T06:53:02Z"},{"alias_kind":"arxiv_version","alias_value":"1912.09860v2","created_at":"2026-07-05T06:53:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1912.09860","created_at":"2026-07-05T06:53:02Z"},{"alias_kind":"pith_short_12","alias_value":"ZIHKZY55756R","created_at":"2026-07-05T06:53:02Z"},{"alias_kind":"pith_short_16","alias_value":"ZIHKZY55756RKZHA","created_at":"2026-07-05T06:53:02Z"},{"alias_kind":"pith_short_8","alias_value":"ZIHKZY55","created_at":"2026-07-05T06:53:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:ZIHKZY55756RKZHA4URAL2HDWR","target":"record","payload":{"canonical_record":{"source":{"id":"1912.09860","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-12-20T14:59:30Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"b4d47fef204efdf600d2eb95c9129eb45091a9e2f55389856b796a19f79599e4","abstract_canon_sha256":"093ae02c459afa46a37eb978f968524028aa705072ebb89ef637f78c55b19724"},"schema_version":"1.0"},"canonical_sha256":"ca0eace3bdff7d1564e0e52205e8e3b44899166e6f712e063fa0f850ebb30364","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:53:02.064023Z","signature_b64":"4ZPuM4WtHkTeI1JGr5VCvrX3B5A44gmVn4u+JOLurbQf71iDqt1xstsZIgA6k80tkSXUPZHK9cvjEbv3eZ0RDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ca0eace3bdff7d1564e0e52205e8e3b44899166e6f712e063fa0f850ebb30364","last_reissued_at":"2026-07-05T06:53:02.063432Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:53:02.063432Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1912.09860","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":"2026-07-05T06:53:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ofS7TWv7sjYSdr+dJpxkeZ/Mu+PHG6IPaiXWmVhQBy0tPqNGyzlC5TtbFDs7gHJolLL2krkjvMCaDptiki39BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T07:18:01.570802Z"},"content_sha256":"ba5a5ede54a682232e198fe5bd76394a9545d3f67c76492880753eb7ce08d96b","schema_version":"1.0","event_id":"sha256:ba5a5ede54a682232e198fe5bd76394a9545d3f67c76492880753eb7ce08d96b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:ZIHKZY55756RKZHA4URAL2HDWR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Hessian matrices, automorphisms of $p$-groups, and torsion points of elliptic curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.GR","authors_text":"Christopher Voll, Mima Stanojkovski","submitted_at":"2019-12-20T14:59:30Z","abstract_excerpt":"We describe the automorphism groups of finite $p$-groups arising naturally via Hessian determinantal representations of elliptic curves defined over number fields. Moreover, we derive explicit formulas for the orders of these automorphism groups for elliptic curves of $j$-invariant $1728$ given in Weierstrass form. We interpret these orders in terms of the numbers of $3$-torsion points (or flex points) of the relevant curves over finite fields. Our work greatly generalizes and conceptualizes previous examples given by du Sautoy and Vaughan-Lee. It explains, in particular, why the orders arisin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1912.09860","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1912.09860/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T06:53:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RPdfWTvLF+b2cdV2LrH+xHSaj6INP/NuzMdszFHTjqB5+iSJi10yDGQtOHDddzzIFL55/NTzykdbjSri7rz2CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T07:18:01.571277Z"},"content_sha256":"c082d4d3984761196b45b93eae3628c824f188a8e9a4dab3036558613b5a4765","schema_version":"1.0","event_id":"sha256:c082d4d3984761196b45b93eae3628c824f188a8e9a4dab3036558613b5a4765"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZIHKZY55756RKZHA4URAL2HDWR/bundle.json","state_url":"https://pith.science/pith/ZIHKZY55756RKZHA4URAL2HDWR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZIHKZY55756RKZHA4URAL2HDWR/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":"2026-07-07T07:18:01Z","links":{"resolver":"https://pith.science/pith/ZIHKZY55756RKZHA4URAL2HDWR","bundle":"https://pith.science/pith/ZIHKZY55756RKZHA4URAL2HDWR/bundle.json","state":"https://pith.science/pith/ZIHKZY55756RKZHA4URAL2HDWR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZIHKZY55756RKZHA4URAL2HDWR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:ZIHKZY55756RKZHA4URAL2HDWR","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":"093ae02c459afa46a37eb978f968524028aa705072ebb89ef637f78c55b19724","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-12-20T14:59:30Z","title_canon_sha256":"b4d47fef204efdf600d2eb95c9129eb45091a9e2f55389856b796a19f79599e4"},"schema_version":"1.0","source":{"id":"1912.09860","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1912.09860","created_at":"2026-07-05T06:53:02Z"},{"alias_kind":"arxiv_version","alias_value":"1912.09860v2","created_at":"2026-07-05T06:53:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1912.09860","created_at":"2026-07-05T06:53:02Z"},{"alias_kind":"pith_short_12","alias_value":"ZIHKZY55756R","created_at":"2026-07-05T06:53:02Z"},{"alias_kind":"pith_short_16","alias_value":"ZIHKZY55756RKZHA","created_at":"2026-07-05T06:53:02Z"},{"alias_kind":"pith_short_8","alias_value":"ZIHKZY55","created_at":"2026-07-05T06:53:02Z"}],"graph_snapshots":[{"event_id":"sha256:c082d4d3984761196b45b93eae3628c824f188a8e9a4dab3036558613b5a4765","target":"graph","created_at":"2026-07-05T06:53:02Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1912.09860/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We describe the automorphism groups of finite $p$-groups arising naturally via Hessian determinantal representations of elliptic curves defined over number fields. Moreover, we derive explicit formulas for the orders of these automorphism groups for elliptic curves of $j$-invariant $1728$ given in Weierstrass form. We interpret these orders in terms of the numbers of $3$-torsion points (or flex points) of the relevant curves over finite fields. Our work greatly generalizes and conceptualizes previous examples given by du Sautoy and Vaughan-Lee. It explains, in particular, why the orders arisin","authors_text":"Christopher Voll, Mima Stanojkovski","cross_cats":["math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-12-20T14:59:30Z","title":"Hessian matrices, automorphisms of $p$-groups, and torsion points of elliptic curves"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1912.09860","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:ba5a5ede54a682232e198fe5bd76394a9545d3f67c76492880753eb7ce08d96b","target":"record","created_at":"2026-07-05T06:53:02Z","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":"093ae02c459afa46a37eb978f968524028aa705072ebb89ef637f78c55b19724","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-12-20T14:59:30Z","title_canon_sha256":"b4d47fef204efdf600d2eb95c9129eb45091a9e2f55389856b796a19f79599e4"},"schema_version":"1.0","source":{"id":"1912.09860","kind":"arxiv","version":2}},"canonical_sha256":"ca0eace3bdff7d1564e0e52205e8e3b44899166e6f712e063fa0f850ebb30364","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ca0eace3bdff7d1564e0e52205e8e3b44899166e6f712e063fa0f850ebb30364","first_computed_at":"2026-07-05T06:53:02.063432Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:53:02.063432Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4ZPuM4WtHkTeI1JGr5VCvrX3B5A44gmVn4u+JOLurbQf71iDqt1xstsZIgA6k80tkSXUPZHK9cvjEbv3eZ0RDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T06:53:02.064023Z","signed_message":"canonical_sha256_bytes"},"source_id":"1912.09860","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ba5a5ede54a682232e198fe5bd76394a9545d3f67c76492880753eb7ce08d96b","sha256:c082d4d3984761196b45b93eae3628c824f188a8e9a4dab3036558613b5a4765"],"state_sha256":"ad14a0d84b8ae56756346013e55d8323abc571cbd1bcbb3a1dc4419a40c9e1db"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5vFj8BihkrcYN12NzTbRI3Yx/EbcSvdj3f8fXGUJrwYvBUvi6lksZjXgCZbxNX09upkj9+oltlgz2q+9XOuxAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-07T07:18:01.577334Z","bundle_sha256":"4048ef0e06971b5155dfcbc301d8bd9d55c1b6bfbe17324991abf7331b395dc6"}}