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Statistics of the Genus Number of $S_3 \times C_2$ and $D_4$ -fields

This paper establishes statistical results and precise moment formulas for the genus numbers of $S_3 \times C_q$ , $D_4$ , and pure quartic fields, while proposing a heuristic conjecture regarding the families where the genus density is zero.

Original authors: Anup B. Dixit, Sunil Kumar Pasupulati

Published 2026-05-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Anup B. Dixit, Sunil Kumar Pasupulati

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to understand the hidden "fingerprint" of a number system. In the world of mathematics, these systems are called number fields. Every number field has a specific "fingerprint" called its genus number.

Think of the genus number as a measure of how "messy" or "complicated" the rules of arithmetic are in that specific world. It tells you how much the system has been disturbed by "ramification"—a fancy word for when prime numbers (the building blocks of math) behave strangely or get tangled up when you move from the standard number line into this new world.

This paper, written by Anup B. Dixit and Sunil Kumar Pasupulati, is essentially a massive census. The authors went out and counted how many number fields have specific genus numbers (1, 2, 4, etc.) within several different families of these mathematical worlds. They wanted to answer a big question: Is having a specific fingerprint common, or is it a rare anomaly?

Here is a breakdown of their findings using simple analogies:

1. The Two Types of Families

The authors studied three main groups of number fields. Their results split these groups into two very different behaviors:

Group A: The "Common" Families ( $S_3 \times C_2$ , $S_3 \times C_q$ , and $D_4$ )

Imagine a huge city where people are born with different hair colors. In this city, if you pick a random person, there is a real, non-zero chance they will have a specific hair color (like red hair).

The Finding: For these families, the authors proved that for any "allowed" genus number (like 1, 2, or 3), a positive proportion of the fields will have that number.
The Analogy: If you walk into a room full of these specific number fields, you won't just find one or two with a genus number of 1; you will find a whole crowd of them. In fact, about 45% of the $S_3 \times C_2$ fields they studied have the simplest possible fingerprint (genus number 1).
The Result: They calculated exactly how many fields have each fingerprint. For example, they found that fields with a genus number of 1 are the most common, but fields with genus numbers like 2, 3, or 4 are also plentiful enough to be statistically significant.

Group B: The "Rare" Families (Pure Quartic Fields)

Now, imagine a different city where, theoretically, people could have red hair, but in reality, almost nobody does. If you look at a million people, you might find one or two, but if you look at a billion, the percentage of redheads is still effectively zero.

The Finding: For "Pure Quartic Fields" (a specific, stricter type of number field), the authors proved that for any fixed genus number, the percentage of fields having that number is zero.
The Analogy: It's like looking for a specific, rare coin in a massive pile of sand. You might find one if you dig for a long time, but as the pile of sand gets infinitely bigger, the chance of finding that specific coin in a random handful becomes zero.
The Result: Even though these fields exist, they are so rare that they don't form a "positive proportion" of the total. They are statistical outliers.

2. The Big Conjecture: The "Abelian" Rule

The authors noticed a pattern and made a bold guess (a conjecture) to explain why some families are common and others are rare.

The Rule: They suggest that if a family of number fields is "close" to being Abelian (a mathematical term meaning the rules of the system are very orderly and predictable, like a well-organized library), then having a specific genus number will be a zero-density event (the "rare coin" scenario).
The Logic: The more orderly and "Abelian" the family is, the more likely it is that the "messiness" (genus number) will vary wildly, making any single specific value extremely rare.
The Evidence: They tested this idea on two other complex families (Octic $D_4$ fields and $C_7 \rtimes C_3$ fields) and found that the math supports their guess: these orderly families seem to have zero density for fixed genus numbers.

3. What They Actually Did

The paper is heavy on calculation, but the core tasks were:

Counting: They used advanced math to count how many fields exist up to a certain size (discriminant) for specific genus numbers.
Formula Making: They created precise formulas (like the ones in Theorem 1.1 and 1.3) that tell you exactly how many fields you will find with a specific fingerprint as the pool of fields gets larger.
Averages: They calculated the "average" genus number for these families, showing that while the average might be low, the distribution is spread out in a predictable way for the "Common" families.

Summary

For some families ( $S_3 \times C_2$ , etc.): Having a specific genus number is common. You can expect to find a healthy percentage of fields with that number.
For other families (Pure Quartic, and likely others close to "Abelian"): Having a specific genus number is extremely rare (zero density). You won't find a "typical" number; the values will keep changing as you look at larger and larger sets.

The paper doesn't claim this helps with cryptography, physics, or engineering right now. It is purely a study of the "population statistics" of these abstract mathematical worlds, trying to understand the rules that govern how often certain mathematical properties appear.

Technical Summary: Statistics of the Genus Number of $S_3 \times C_2$ and $D_4$ -Fields

Problem Statement
The paper investigates the statistical distribution of the genus number $g_K$ across various families of number fields ordered by discriminant. The genus number, defined as the degree of the genus field $K^*$ over $K$ , measures the contribution of ramification to the ideal class group. While the distribution of genus numbers is well-understood for quadratic fields (where fixed genus numbers occur with density zero) and certain cubic and quintic fields (where specific genus values occur with positive density), a general dichotomy remains unclear. The central problem is to determine which families of number fields exhibit a "positive proportion" phenomenon for fixed genus numbers and which exhibit a "zero-density" phenomenon.

Methodology
The authors employ analytic number theory techniques, specifically asymptotic counting of number fields with prescribed local ramification conditions. Key methodological components include:

Genus Theory Decomposition: Utilizing results by Ishida, Gauss, and Fröhlich to decompose the genus field and express the genus number in terms of ramified primes and their local behaviors (e.g., tame vs. wild ramification).
Compositum Analysis: For fields of the form $L = KF$ (where $K$ and $F$ are linearly disjoint), the genus field is shown to be the compositum of the individual genus fields ( $L^* = K^*F^*$ ), allowing the genus number to be factored.
Counting Formulas: The paper relies on asymptotic formulas for the number of cubic fields (Taniguchi-Thorne) and $D_4$ -fields (Cohen-Diaz y Diaz-Olivier, refined by McGown-Tucker) with specific ramification constraints.
Parametrization of Extensions: For $D_4$ -fields, the authors utilize the parametrization of quadratic extensions over a quadratic base field via the 2-Selmer group and ray class groups to count fields with specific relative discriminants.
Heuristic Arguments: In the final section, the authors use heuristic reasoning based on the distribution of prime divisors in discriminants to support a conjecture regarding zero-density families.

Key Contributions and Results

1. $S_3 \times C_2$ Fields (Sextic Fields)
The authors establish precise asymptotic formulas for the number of $S_3 \times C_2$ fields with a fixed genus number $g_K = 2^k 3^\ell$ .

Theorem 1.1: For integers $k, \ell \ge 0$ , the number of such fields with discriminant $\le X$ is given by $|Q^2_{k,\ell}(X)| = A(k)B(\ell)\zeta(2)^{-1}\sqrt{X} + O(X^{8/17+\epsilon})$ .
Significance: This result demonstrates that every admissible genus number is attained by a positive proportion of $S_3 \times C_2$ fields. The paper provides explicit constants $A(k)$ and $B(\ell)$ and a table of numerical proportions (e.g., $\approx 45\%$ of such fields have genus number 1).

2. $S_3 \times C_q$ Fields ( $q \ge 5$ Prime)
Extending the analysis to $S_3 \times C_q$ fields, the genus number takes the form $g_K = 3^\ell q^k$ .

Theorem 1.2: The authors prove that for any prime $q \ge 5$ and non-negative integers $k, \ell$ , the count of such fields is $|Q^q_{k,\ell}(X)| = C_q(k, \ell) X^{1/q} + O(X^{16/17q})$ .
Significance: Similar to the $q=2$ case, this confirms a positive proportion of fields attain any admissible genus number in this family.

3. $D_4$ -Fields (Quartic Fields)
The paper analyzes quartic fields with Galois closure $D_4$ .

Theorem 1.3: For every $a \in \mathbb{Z}_{\ge 0}$ , the number of $D_4$ -fields with genus number $2^a$ and discriminant $\le X$ is asymptotic to $\delta_a X$ , where $\delta_a > 0$ .
Significance: This establishes that, like the $S_3 \times C_q$ families, the $D_4$ -field family exhibits a positive proportion phenomenon for all admissible genus numbers.

4. Pure Quartic Fields
The authors investigate a subfamily of $D_4$ -fields: pure quartic fields $K = \mathbb{Q}(\sqrt[4]{a})$ .

Theorem 1.5: In contrast to the general $D_4$ family, for any fixed genus number $2^a$ , the proportion of pure quartic fields attaining this genus number tends to zero as $X \to \infty$ .
Significance: This identifies a specific family where the zero-density phenomenon occurs, despite the fields being a subfamily of those with positive density.

5. Moments and Conjectures

Moments: The paper derives formulas for the mean and higher moments of the genus number over $S_3 \times C_q$ fields (Section 7), showing that the distribution is heavily skewed away from 1 for higher moments.
Conjecture 1.4: Based on the dichotomy observed (positive density for "non-abelian-like" extensions vs. zero density for "abelian-like" or pure fields), the authors conjecture that for any finite group $G$ , if a family of fields admits a tower of abelian extensions, the density of fields with any fixed genus number is zero.
Heuristic Support: Section 8 provides heuristic evidence for this conjecture using octic $D_4$ -fields and fields with Galois group $C_7 \rtimes C_3$ , arguing that the constraint of avoiding certain congruence classes of ramified primes forces the density to zero.

Significance of the Work
The paper provides a comprehensive statistical picture of genus numbers across several major families of number fields. Its primary contribution is the rigorous proof that the "positive proportion" phenomenon, previously observed in cubic and quintic fields, extends to $S_3 \times C_q$ and general $D_4$ -fields. Conversely, it highlights that this behavior is not universal, demonstrating the zero-density phenomenon in pure quartic fields. The proposed conjecture (1.4) offers a potential unifying principle linking the structure of the Galois group (specifically the presence of abelian towers) to the statistical distribution of genus numbers, suggesting that families "close to being abelian" are the ones where fixed genus numbers become rare.