{"paper":{"title":"Statistics of the Genus Number of $S_3 \\times C_q$ and $D_4$-fields","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Genus numbers of S3×Cq-fields have explicit averages and moments, with analogous statistics for D4 and pure quartic fields.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Anup B. Dixit, Sunil Kumar Pasupulati","submitted_at":"2026-05-06T11:41:40Z","abstract_excerpt":"The genus number of a number field is a fundamental invariant which measures the contribution of ramification to its ideal class group. In this paper, we establish the statistics for the genus number for $S_3\\times C_q$-fields for $q\\neq 3$ a prime number, $D_4$-fields and pure quartic fields. We also obtain precise results on the average and higher moments of the genus distribution within the family of $S_3\\times C_q$-fields. Finally, based on heuristics, we formulate a conjecture identifying families for which one should expect the genus density to be zero, i.e., only a density zero subset o"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We establish the statistics for the genus number for S3×Cq-fields for q≠3 a prime number, D4-fields and pure quartic fields. We also obtain precise results on the average and higher moments of the genus distribution within the family of S3×Cq-fields.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The conjecture on families with genus density zero rests on unspecified heuristics that predict the vanishing of the density of fields attaining any fixed genus number; if these heuristics fail for the ramification or Galois conditions in the families, the conjecture does not follow.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Establishes statistics, averages, and moments of genus numbers for S3×Cq, D4, and pure quartic fields, plus a conjecture on zero-density families.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Genus numbers of S3×Cq-fields have explicit averages and moments, with analogous statistics for D4 and pure quartic fields.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"7eade35cc362b75b9f38839681d3188560dbdecf1c9a0db88d5cd5391b82a3f4"},"source":{"id":"2605.04792","kind":"arxiv","version":2},"verdict":{"id":"49ae5feb-0fa9-430c-8b62-5505885cae47","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T17:46:24.592048Z","strongest_claim":"We establish the statistics for the genus number for S3×Cq-fields for q≠3 a prime number, D4-fields and pure quartic fields. We also obtain precise results on the average and higher moments of the genus distribution within the family of S3×Cq-fields.","one_line_summary":"Establishes statistics, averages, and moments of genus numbers for S3×Cq, D4, and pure quartic fields, plus a conjecture on zero-density families.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The conjecture on families with genus density zero rests on unspecified heuristics that predict the vanishing of the density of fields attaining any fixed genus number; if these heuristics fail for the ramification or Galois conditions in the families, the conjecture does not follow.","pith_extraction_headline":"Genus numbers of S3×Cq-fields have explicit averages and moments, with analogous statistics for D4 and pure quartic fields."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.04792/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T22:01:29.097911Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T14:10:05.762464Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"53f133cee0952c8bb263528c33bad6fae0387d56082c9b5be762dfae035d4344"},"references":{"count":20,"sample":[{"doi":"","year":2018,"title":"Kübra Benli, On the number of pure fields of prime degree,Colloq. Math.,153(2018), no.1, 39-50","work_id":"623c9b76-04df-4868-8398-970669535a09","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2007,"title":"Manjul Bhargava, Mass formulae for extensions of local fields, and conjectures on the density of number field discriminants,Int. Math. Res. Not., (2007), no.17, 1-20","work_id":"eacf7f14-ead2-4142-a7c7-6876322e78e9","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2002,"title":"Henri Cohen, Francisco Diaz y Diaz, and Michel Olivier, Enumerating quartic dihedral extensions ofQ, Compos. Math.,133(2002), no.1, 65-93. STATISTICS OF THE GENUS NUMBER OFS3 ×C q ANDD 4-FIELDS 29","work_id":"0c9b0bde-53f7-4bd9-bc97-235c709ad783","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"Christopher Frei, Daniel Loughran, and Rachel Newton, Distribution of genus numbers of abelian number fields,J. Lond. Math. Soc.(2),107(2023), no.6, 2197-2217","work_id":"582e7539-920f-4ccf-acdc-e70128757662","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1959,"title":"Fröhlich, The genus field and genus group in finite number fields","work_id":"7def569b-2705-44de-b1c1-78492db47a41","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":20,"snapshot_sha256":"f85893b4a64159e7ed1b802e38c1da688ef600617b22ba48ed4884ee601e8fd0","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"25c593c6d55d0a5846ea9c7bb5d7a06c3c87d6d3380f68c491acb13aac5e2201"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}