{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:TUSLAJ62MDEGDYUBL3RNRV3QJE","short_pith_number":"pith:TUSLAJ62","canonical_record":{"source":{"id":"2005.07329","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-05-15T02:31:31Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"268e402f749582d87bc68f1c7dd86fbcfc54a962e99aa3afbaf3d1a38ea6a169","abstract_canon_sha256":"12d26d2ac6e5471a9f70a203ba002b938a38cf58b2523cfebcaa30b86fce32bb"},"schema_version":"1.0"},"canonical_sha256":"9d24b027da60c861e2815ee2d8d77049318315e76a7caad32fb1a5a5ed6908e8","source":{"kind":"arxiv","id":"2005.07329","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2005.07329","created_at":"2026-07-05T10:52:07Z"},{"alias_kind":"arxiv_version","alias_value":"2005.07329v2","created_at":"2026-07-05T10:52:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2005.07329","created_at":"2026-07-05T10:52:07Z"},{"alias_kind":"pith_short_12","alias_value":"TUSLAJ62MDEG","created_at":"2026-07-05T10:52:07Z"},{"alias_kind":"pith_short_16","alias_value":"TUSLAJ62MDEGDYUB","created_at":"2026-07-05T10:52:07Z"},{"alias_kind":"pith_short_8","alias_value":"TUSLAJ62","created_at":"2026-07-05T10:52:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:TUSLAJ62MDEGDYUBL3RNRV3QJE","target":"record","payload":{"canonical_record":{"source":{"id":"2005.07329","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-05-15T02:31:31Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"268e402f749582d87bc68f1c7dd86fbcfc54a962e99aa3afbaf3d1a38ea6a169","abstract_canon_sha256":"12d26d2ac6e5471a9f70a203ba002b938a38cf58b2523cfebcaa30b86fce32bb"},"schema_version":"1.0"},"canonical_sha256":"9d24b027da60c861e2815ee2d8d77049318315e76a7caad32fb1a5a5ed6908e8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:52:07.346514Z","signature_b64":"pi7RdB8FvgC+NDyIPDd3dfNL/xrN8tjXQrv/YVdkpVJ26rr4DAW14Jeb406S2EASPpSrQoAhcHsMQcY0bfV/Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9d24b027da60c861e2815ee2d8d77049318315e76a7caad32fb1a5a5ed6908e8","last_reissued_at":"2026-07-05T10:52:07.345862Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:52:07.345862Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2005.07329","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":"2026-07-05T10:52:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Esdkw3WzBebWJ/gHiTBGAE9jjNK7OEzBcMz8np3RhvsX5Sa5TuChgc0H29bAINw9wEIIxbnEwpVcWx99cQcEDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T14:38:13.027732Z"},"content_sha256":"f4493576a1428c6d3f337a737bedbeef32973e45194b6c51522175b9bad07058","schema_version":"1.0","event_id":"sha256:f4493576a1428c6d3f337a737bedbeef32973e45194b6c51522175b9bad07058"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:TUSLAJ62MDEGDYUBL3RNRV3QJE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Presentations of Galois groups of maximal extensions with restricted ramification","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.NT","authors_text":"Yuan Liu","submitted_at":"2020-05-15T02:31:31Z","abstract_excerpt":"Motivated by the work of Lubotzky, we use Galois cohomology to study the difference between the number of generators and the minimal number of relations in a presentation of the Galois group $G_S(k)$ of the maximal extension of a global field $k$ that is unramified outside a finite set $S$ of places, as $k$ varies among a certain family of extensions of a fixed global field $Q$. We prove a generalized version of the global Euler-Poincar\\'{e} Characteristic, and define a group $B_S(k,A)$, for each finite simple $G_S(k)$-module $A$, to generalize the work of Koch about the pro-$\\ell$ completion "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2005.07329","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/2005.07329/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":"2026-07-05T10:52:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RI//5jDUGGb1rm6SmtuF1hKAo9dNIdSxjbtMRWabC91mXwo2c95tiR+bA6VLsDkoZIHmx/yJGKLf0uDbmMESDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T14:38:13.028256Z"},"content_sha256":"133a346f4f466534660c75336a729b46f809b8e9bd1f476d2136007c631ff7ae","schema_version":"1.0","event_id":"sha256:133a346f4f466534660c75336a729b46f809b8e9bd1f476d2136007c631ff7ae"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TUSLAJ62MDEGDYUBL3RNRV3QJE/bundle.json","state_url":"https://pith.science/pith/TUSLAJ62MDEGDYUBL3RNRV3QJE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TUSLAJ62MDEGDYUBL3RNRV3QJE/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":"2026-08-18T14:38:13Z","links":{"resolver":"https://pith.science/pith/TUSLAJ62MDEGDYUBL3RNRV3QJE","bundle":"https://pith.science/pith/TUSLAJ62MDEGDYUBL3RNRV3QJE/bundle.json","state":"https://pith.science/pith/TUSLAJ62MDEGDYUBL3RNRV3QJE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TUSLAJ62MDEGDYUBL3RNRV3QJE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:TUSLAJ62MDEGDYUBL3RNRV3QJE","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":"12d26d2ac6e5471a9f70a203ba002b938a38cf58b2523cfebcaa30b86fce32bb","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-05-15T02:31:31Z","title_canon_sha256":"268e402f749582d87bc68f1c7dd86fbcfc54a962e99aa3afbaf3d1a38ea6a169"},"schema_version":"1.0","source":{"id":"2005.07329","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2005.07329","created_at":"2026-07-05T10:52:07Z"},{"alias_kind":"arxiv_version","alias_value":"2005.07329v2","created_at":"2026-07-05T10:52:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2005.07329","created_at":"2026-07-05T10:52:07Z"},{"alias_kind":"pith_short_12","alias_value":"TUSLAJ62MDEG","created_at":"2026-07-05T10:52:07Z"},{"alias_kind":"pith_short_16","alias_value":"TUSLAJ62MDEGDYUB","created_at":"2026-07-05T10:52:07Z"},{"alias_kind":"pith_short_8","alias_value":"TUSLAJ62","created_at":"2026-07-05T10:52:07Z"}],"graph_snapshots":[{"event_id":"sha256:133a346f4f466534660c75336a729b46f809b8e9bd1f476d2136007c631ff7ae","target":"graph","created_at":"2026-07-05T10:52:07Z","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/2005.07329/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Motivated by the work of Lubotzky, we use Galois cohomology to study the difference between the number of generators and the minimal number of relations in a presentation of the Galois group $G_S(k)$ of the maximal extension of a global field $k$ that is unramified outside a finite set $S$ of places, as $k$ varies among a certain family of extensions of a fixed global field $Q$. We prove a generalized version of the global Euler-Poincar\\'{e} Characteristic, and define a group $B_S(k,A)$, for each finite simple $G_S(k)$-module $A$, to generalize the work of Koch about the pro-$\\ell$ completion ","authors_text":"Yuan Liu","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-05-15T02:31:31Z","title":"Presentations of Galois groups of maximal extensions with restricted ramification"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2005.07329","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:f4493576a1428c6d3f337a737bedbeef32973e45194b6c51522175b9bad07058","target":"record","created_at":"2026-07-05T10:52:07Z","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":"12d26d2ac6e5471a9f70a203ba002b938a38cf58b2523cfebcaa30b86fce32bb","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-05-15T02:31:31Z","title_canon_sha256":"268e402f749582d87bc68f1c7dd86fbcfc54a962e99aa3afbaf3d1a38ea6a169"},"schema_version":"1.0","source":{"id":"2005.07329","kind":"arxiv","version":2}},"canonical_sha256":"9d24b027da60c861e2815ee2d8d77049318315e76a7caad32fb1a5a5ed6908e8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9d24b027da60c861e2815ee2d8d77049318315e76a7caad32fb1a5a5ed6908e8","first_computed_at":"2026-07-05T10:52:07.345862Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:52:07.345862Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pi7RdB8FvgC+NDyIPDd3dfNL/xrN8tjXQrv/YVdkpVJ26rr4DAW14Jeb406S2EASPpSrQoAhcHsMQcY0bfV/Ag==","signature_status":"signed_v1","signed_at":"2026-07-05T10:52:07.346514Z","signed_message":"canonical_sha256_bytes"},"source_id":"2005.07329","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f4493576a1428c6d3f337a737bedbeef32973e45194b6c51522175b9bad07058","sha256:133a346f4f466534660c75336a729b46f809b8e9bd1f476d2136007c631ff7ae"],"state_sha256":"336bf6a199f30d79cb089f8f9eb04e5de2f6d417b358bfe96b1aaa3998b10282"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2G7+QCTMuWw7ucdLxoOB2COlTA0T3YjW6yCY9aZY7BLVSJrLmrt7OPWocfs9ZlhjQCVlNWjFeqthBab7NiSBCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-18T14:38:13.032451Z","bundle_sha256":"fa6da93c59f9299df9e382e2acd7553c10313fe3291806f4abcbdbcd77e2f8b6"}}