{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2020:TUSLAJ62MDEGDYUBL3RNRV3QJE","short_pith_number":"pith:TUSLAJ62","schema_version":"1.0","canonical_sha256":"9d24b027da60c861e2815ee2d8d77049318315e76a7caad32fb1a5a5ed6908e8","source":{"kind":"arxiv","id":"2005.07329","version":2},"attestation_state":"computed","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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2005.07329","created_at":"2026-07-05T10:52:07.345954+00:00"},{"alias_kind":"arxiv_version","alias_value":"2005.07329v2","created_at":"2026-07-05T10:52:07.345954+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2005.07329","created_at":"2026-07-05T10:52:07.345954+00:00"},{"alias_kind":"pith_short_12","alias_value":"TUSLAJ62MDEG","created_at":"2026-07-05T10:52:07.345954+00:00"},{"alias_kind":"pith_short_16","alias_value":"TUSLAJ62MDEGDYUB","created_at":"2026-07-05T10:52:07.345954+00:00"},{"alias_kind":"pith_short_8","alias_value":"TUSLAJ62","created_at":"2026-07-05T10:52:07.345954+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2411.19318","citing_title":"On the Distribution of Class Groups of Abelian Extensions","ref_index":1998,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TUSLAJ62MDEGDYUBL3RNRV3QJE","json":"https://pith.science/pith/TUSLAJ62MDEGDYUBL3RNRV3QJE.json","graph_json":"https://pith.science/api/pith-number/TUSLAJ62MDEGDYUBL3RNRV3QJE/graph.json","events_json":"https://pith.science/api/pith-number/TUSLAJ62MDEGDYUBL3RNRV3QJE/events.json","paper":"https://pith.science/paper/TUSLAJ62"},"agent_actions":{"view_html":"https://pith.science/pith/TUSLAJ62MDEGDYUBL3RNRV3QJE","download_json":"https://pith.science/pith/TUSLAJ62MDEGDYUBL3RNRV3QJE.json","view_paper":"https://pith.science/paper/TUSLAJ62","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2005.07329&json=true","fetch_graph":"https://pith.science/api/pith-number/TUSLAJ62MDEGDYUBL3RNRV3QJE/graph.json","fetch_events":"https://pith.science/api/pith-number/TUSLAJ62MDEGDYUBL3RNRV3QJE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TUSLAJ62MDEGDYUBL3RNRV3QJE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TUSLAJ62MDEGDYUBL3RNRV3QJE/action/storage_attestation","attest_author":"https://pith.science/pith/TUSLAJ62MDEGDYUBL3RNRV3QJE/action/author_attestation","sign_citation":"https://pith.science/pith/TUSLAJ62MDEGDYUBL3RNRV3QJE/action/citation_signature","submit_replication":"https://pith.science/pith/TUSLAJ62MDEGDYUBL3RNRV3QJE/action/replication_record"}},"created_at":"2026-07-05T10:52:07.345954+00:00","updated_at":"2026-07-05T10:52:07.345954+00:00"}