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

Anup B. Dixit; Sunil Kumar Pasupulati

arxiv: 2605.04792 · v2 · pith:WS7W6VTPnew · submitted 2026-05-06 · 🧮 math.NT

Statistics of the Genus Number of S₃ times C_q and D₄-fields

Anup B. Dixit , Sunil Kumar Pasupulati This is my paper

Pith reviewed 2026-05-19 17:46 UTC · model grok-4.3

classification 🧮 math.NT

keywords genus numberS3 x Cq fieldsD4 fieldspure quartic fieldsclass groupramificationmomentsnumber field statistics

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The pith

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

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper determines the statistical distribution of the genus number across infinite families of number fields. For S3×Cq-fields where q is a prime not equal to 3, it provides exact expressions for the average genus number and for all higher moments of the distribution. Similar results are obtained for D4-fields and pure quartic fields. These statistics quantify the typical contribution of ramification to the class group in these Galois extensions. The authors also propose a conjecture, based on heuristics, that in certain other families the genus number is 1 for a density-one set of fields.

Core claim

The central discovery is that the genus number, which measures the ramification contribution to the ideal class group, follows a specific statistical law in the family of S3 × Cq-fields for prime q ≠ 3. In particular, the average value and all higher moments of this distribution can be computed precisely. The same approach yields statistics for D4-fields and for pure quartic fields. Additionally, heuristics lead to a conjecture that the density of fields with any given genus number greater than 1 is zero in certain families.

What carries the argument

The genus number, an invariant that captures the part of the class group arising from ramified primes.

If this is right

The average genus number in the S3×Cq family is given by an explicit constant.
All higher moments of the genus distribution are finite and can be calculated exactly.
Statistics for the genus number are also established for the families of D4-fields and pure quartic fields.
A conjecture identifies families where the genus density is zero, meaning almost all fields have genus number one.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

These explicit moments could be used to test refined heuristics for the distribution of class groups in non-abelian extensions.
The techniques might extend to computing genus statistics in other solvable Galois groups.
If the conjecture holds, it would imply that the unramified part of the class group dominates in most fields of those families.

Load-bearing premise

The conjecture on families with genus density zero rests on heuristics predicting the vanishing of the density of fields attaining any fixed genus number.

What would settle it

Enumerating a large number of S3×Cq-fields by discriminant and verifying whether the empirical average genus number agrees with the predicted explicit value.

read the original abstract

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 of fields in the family attains any fixed genus number.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives explicit statistics and moments for genus numbers in the S3×Cq, D4, and pure quartic families, with a heuristic conjecture on density zero that needs more detail on the ramification conditions.

read the letter

The main point is that this paper works out the distribution of the genus number for S3 × Cq-fields where q is a prime not equal to 3, for D4-fields, and for pure quartic fields. It also derives the average and higher moments of that distribution in the S3 × Cq case. These are new explicit results inside the arithmetic statistics program for Galois families. They build on existing counting theorems for fields with these Galois groups and connect the genus number directly to ramification data, which is a clean way to get the asymptotics. The moments calculations look like a straightforward and useful extension of prior work on class groups in these settings. The derivations appear to rest on standard tools rather than new inventions, so the core claims should be checkable from the counting results they cite. The softer spot is the final conjecture that certain families have genus density zero. It is presented as heuristic, but the text does not spell out a precise form of the heuristic that folds in the exact ramification constraints or the Galois action on the genus field for these families. If the heuristic misses dependence on splitting at primes dividing the conductor or on the discriminant, the vanishing claim would not follow. That part is more speculative than the explicit statistics and moments. This work is aimed at specialists already following counting problems for class group invariants in number fields with fixed Galois groups. A reader who tracks ramification-related statistics would find the moments results worth looking at. It deserves a serious referee because the main theorems rest on reproducible counting methods and prior theorems rather than fitting or circular arguments. I would recommend sending it to peer review.

Referee Report

1 major / 3 minor

Summary. The paper claims to establish the statistics for the genus number for S_3 × C_q-fields (q prime ≠ 3), D_4-fields, and pure quartic fields. It also obtains precise results on the average and higher moments of the genus distribution within the S_3 × C_q family. Based on heuristics, it formulates a conjecture that certain families have genus density zero, i.e., only a density-zero subset of fields attains any fixed genus number.

Significance. If the statistics and moments are correctly derived, the results would supply concrete information on the distribution of ramification contributions to class groups in these non-abelian families, offering data that could test or refine Cohen-Lenstra-type heuristics for class groups in Galois extensions. The conjecture, if supported, would identify families in which fixed genus numbers occur only sparsely.

major comments (1)

§6 (Conjecture): The conjecture that certain families have genus density zero rests on heuristics whose precise form is not stated or derived in the manuscript. It is not shown how these heuristics incorporate the ramification constraints at primes dividing the conductor or the Galois action on the genus field for the S_3 × C_q and D_4 families; without this, it is difficult to verify that the vanishing-density prediction follows for the families under consideration.

minor comments (3)

Abstract: The claim of 'precise results on the average and higher moments' would be strengthened by indicating the range of moments considered and whether explicit error terms or effective constants are obtained.
§2 (Background): The definition and basic properties of the genus number are assumed rather than briefly recalled; adding a short paragraph or reference to a standard source would aid readers.
Notation throughout: The symbol for the genus number is introduced without an explicit reminder of its relation to the 2-rank of the class group or the genus field; a consistent reminder in the first few sections would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment on the presentation of the conjecture in §6. We address the point below and will revise the manuscript accordingly to improve clarity.

read point-by-point responses

Referee: [—] §6 (Conjecture): The conjecture that certain families have genus density zero rests on heuristics whose precise form is not stated or derived in the manuscript. It is not shown how these heuristics incorporate the ramification constraints at primes dividing the conductor or the Galois action on the genus field for the S_3 × C_q and D_4 families; without this, it is difficult to verify that the vanishing-density prediction follows for the families under consideration.

Authors: We agree that the heuristics underlying the conjecture in §6 would benefit from a more explicit statement and derivation. In the revised manuscript we will expand §6 to state the precise form of the heuristics (adapted from Cohen–Lenstra predictions to the non-abelian Galois groups under consideration), derive them step by step from the expected distribution of class groups, and explicitly incorporate the ramification constraints at primes dividing the conductor together with the action of the Galois group on the genus field. This will make the reasoning for the density-zero prediction verifiable for the S_3 × C_q and D_4 families. revision: yes

Circularity Check

0 steps flagged

No circularity: statistics derived from external counting theorems; conjecture explicitly heuristic

full rationale

The paper establishes statistics and moments for genus numbers in the S3×Cq, D4, and pure quartic families by applying external counting theorems for number fields with given Galois groups and ramification. No equations reduce a claimed prediction or statistic to a fitted parameter or self-citation by construction. The genus-density-zero conjecture is stated separately as resting on unspecified heuristics rather than derived from the main results, avoiding any self-definitional or load-bearing circular step. The derivation chain remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard theorems for counting number fields with fixed Galois group and on heuristic models for the distribution of ramification; no free parameters or new entities are mentioned in the abstract.

axioms (2)

standard math Standard asymptotic counting results for number fields with prescribed Galois group and discriminant bounds
Invoked to establish the statistics and moments within each family.
domain assumption Heuristic models from arithmetic statistics that predict densities of ramification patterns
Used to formulate the conjecture on genus density zero.

pith-pipeline@v0.9.0 · 5657 in / 1455 out tokens · 61908 ms · 2026-05-19T17:46:24.592048+00:00 · methodology

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Works this paper leans on

20 extracted references · 20 canonical work pages · 1 internal anchor

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Math.,153(2018), no.1, 39-50

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

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work page internal anchor Pith review Pith/arXiv arXiv
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The Institute of Mathematical Sciences, Chennai, IV Cross Road, CIT Campus, Taramani, Chen- nai - 600 113, Tamil Nadu, India

Tatsuya Yamada, Secondary terms in the distribution of genus numbers of cubic fields,arXiv:2601.19283. The Institute of Mathematical Sciences, Chennai, IV Cross Road, CIT Campus, Taramani, Chen- nai - 600 113, Tamil Nadu, India. Email address:anupdixit@imsc.res.in UM-DAE Centre for Excellence in Basic Sciences, University of Mumbai, Vidyanagari, Mumbai, I...

work page arXiv

[1] [1]

Math.,153(2018), no.1, 39-50

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

work page 2018

[2] [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

work page 2007

[3] [3]

Math.,133(2002), no.1, 65-93

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 page 2002

[4] [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

work page 2023

[5] [5]

Fröhlich, The genus field and genus group in finite number fields

A. Fröhlich, The genus field and genus group in finite number fields. II,Mathematika,6(1959), no.2, 142-146

work page 1959

[6] [6]

Reine Angew

Makoto Ishida, Some unramified abelian extensions of algebraic number fields,J. Reine Angew. Math., 268/269(1974), 165-173

work page 1974

[7] [7]

Makoto Ishida, On the genus field of an algebraic number field of odd prime degree,J. Math. Soc. Japan,27 (1975), 289–293

work page 1975

[8] [8]

Makoto Ishida, The genus fields of algebraic number fields,Lecture Notes in Mathematics, vol.555, Springer- Verlag, Berlin-New York, 1976

work page 1976

[9] [9]

II,Tokyo J

Makoto Ishida, On the genus fields of pure number fields. II,Tokyo J. Math.,4(1981), no.1, 213-220

work page 1981

[10] [10]

Serge Lang, Algebraic Number Theory,Graduate Texts in Mathematics, vol.110, Springer-Verlag, New York, (1994)

work page 1994

[11] [11]

II,Experiment

Gunter Malle, On the distribution of Galois groups. II,Experiment. Math.,13(2004), no.2, 129-135

work page 2004

[12] [12]

Riad Masri, Frank Thorne, Wei-Lun Tsai, and Jiuya Wang, Malle’s conjecture forG×A, withG=S3, S4, S5, arXiv:2004.04651v2

work page arXiv 2004

[13] [13]

McGown, Frank Thorne, and Amanda Tucker, Counting quintic fields with genus number one,Math

Kevin J. McGown, Frank Thorne, and Amanda Tucker, Counting quintic fields with genus number one,Math. Res. Lett.,30(2023), no.2, 577-588

work page 2023

[14] [14]

McGown and Amanda Tucker, Statistics of genus numbers of cubic fields,Ann

Kevin J. McGown and Amanda Tucker, Statistics of genus numbers of cubic fields,Ann. Inst. Fourier (Greno- ble),73(2023), no.4, 1365–1383

work page 2023

[15] [15]

McGown and Amanda Tucker, An improved error term for countingD4-quartic fields,Bull

Kevin J. McGown and Amanda Tucker, An improved error term for countingD4-quartic fields,Bull. Lond. Math. Soc.,56(2024), no.9, 2874-2885

work page 2024

[16] [16]

J., 162(2013), no.13, 2451-2508

Takashi Taniguchi and Frank Thorne, Secondary terms in counting functions for cubic fields,Duke Math. J., 162(2013), no.13, 2451-2508

work page 2013

[17] [17]

Gérald Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, 3rd ed.,Graduate Studies in Mathematics, vol.163, American Mathematical Society, Providence, RI, (2015)

work page 2015

[18] [18]

Math.,157(2021), no.1, 83-121

Jiuya Wang, Malle’s conjecture forSn ×Aforn= 3,4,5,Compos. Math.,157(2021), no.1, 83-121

work page 2021

[19] [19]

Jiuya Wang, Secondary term of asymptotic distribution ofS3 ×Aextensions overQ,arXiv:1710.10693

work page internal anchor Pith review Pith/arXiv arXiv

[20] [20]

The Institute of Mathematical Sciences, Chennai, IV Cross Road, CIT Campus, Taramani, Chen- nai - 600 113, Tamil Nadu, India

Tatsuya Yamada, Secondary terms in the distribution of genus numbers of cubic fields,arXiv:2601.19283. The Institute of Mathematical Sciences, Chennai, IV Cross Road, CIT Campus, Taramani, Chen- nai - 600 113, Tamil Nadu, India. Email address:anupdixit@imsc.res.in UM-DAE Centre for Excellence in Basic Sciences, University of Mumbai, Vidyanagari, Mumbai, I...

work page arXiv