Pith Number

pith:WS7W6VTP

pith:2026:WS7W6VTPF7SNAW3QED5XWDEXYA

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Statistics of the Genus Number of $S_3 \times C_q$ and $D_4$-fields

Anup B. Dixit, Sunil Kumar Pasupulati

Genus numbers of S3×Cq-fields have explicit averages and moments, with analogous statistics for D4 and pure quartic fields.

arxiv:2605.04792 v2 · 2026-05-06 · math.NT

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Claims

C1strongest 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.

C2weakest 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.

C3one 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.

References

20 extracted · 20 resolved · 1 Pith anchors

[1] Kübra Benli, On the number of pure fields of prime degree,Colloq. Math.,153(2018), no.1, 39-50 2018

[2] 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 2007

[3] 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 2002

[4] 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 2023

[5] Fröhlich, The genus field and genus group in finite number fields 1959

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:01:42.757888Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

b4bf6f566f2fe4d05b7020fb7b0c97c00694dfd788b1b04ed0dff798e2102d3a

Aliases

arxiv: 2605.04792 · arxiv_version: 2605.04792v2 · doi: 10.48550/arxiv.2605.04792 · pith_short_12: WS7W6VTPF7SN · pith_short_16: WS7W6VTPF7SNAW3Q · pith_short_8: WS7W6VTP

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/WS7W6VTPF7SNAW3QED5XWDEXYA \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: b4bf6f566f2fe4d05b7020fb7b0c97c00694dfd788b1b04ed0dff798e2102d3a

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "8562962a8b1e8daf814478434e7108152c7ace57f6c0d389bbba2f8e26e105c9",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-05-06T11:41:40Z",
    "title_canon_sha256": "48adac45454b70b5d42efe126066e261d98a016e745cc52c2b9f197515620238"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.04792",
    "kind": "arxiv",
    "version": 2
  }
}