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Infinite Moments of Class Groups for Solvable Fields with a Normal Abelian Subgroup

This paper establishes an upper bound on the number of solutions to restricted ramification problems for solvable Galois groups using class field theory and the Minkowski bound, ultimately proving that the $\mathbb{Z}/3\mathbb{Z}$ -moment of class groups is infinite for non-Galois cubic fields ordered by the product of ramified primes.

Original authors: Weitong Wang

Published 2026-06-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Weitong Wang

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 vast collection of secret societies. In the world of mathematics, these societies are called number fields, and their fingerprints are called class groups.

This paper, written by Weitong Wang, is a deep dive into how these fingerprints are distributed. Specifically, it asks: "If we look at a huge crowd of these number fields, how often do we see a specific type of fingerprint?"

Here is the breakdown of the paper's journey, using simple analogies.

1. The Setup: Counting Secret Societies

Imagine you have a giant library of books (number fields). Usually, mathematicians organize these books by their "size" (called the discriminant).

The Goal: The author wants to count how many books have a specific "fingerprint" (a specific structure in their class group).
The Problem: Sometimes, when you look for a specific fingerprint, you find it never happens (probability is zero), or you find it happens so often that the average count explodes to infinity. This is called the "infinite moment" phenomenon.

2. The New Sorting Method: The "Ramified Primes" Tag

In the past, people sorted these books by their total size. But this paper introduces a new way to sort them: by the product of their "ramified primes."

The Analogy: Imagine every book has a list of "stamps" on it. Some stamps are common, some are rare. "Ramified primes" are like special, heavy stamps that stick to the book.
Instead of weighing the whole book, the author decides to sort the library based on the combined weight of just these heavy stamps.
The paper argues that if you sort the library this way, you can prove something very surprising about the fingerprints.

3. The Main Discovery: The "Infinite Moment"

The paper focuses on a specific type of group structure called solvable groups (think of these as societies with a clear chain of command, like a military or a corporate hierarchy).

The author proves a "Zero-Probability, Infinite-Moment" rule:

Zero Probability: If you pick a random book from this specific library, the chance that its fingerprint is a "small" version (specifically, having a small number of certain features) is zero. It's like trying to find a perfectly round square; it just doesn't happen in this context.
Infinite Moment: However, if you try to calculate the average number of these features across the whole library, the number blows up to infinity.

The Metaphor: Imagine a room full of people. If you ask, "What is the chance that a random person has exactly 0 hairs on their head?" the answer might be zero. But if you ask, "What is the average number of hairs?" and the room is filled with people who have millions of hairs, the average becomes infinite. The distribution is so skewed that the "average" breaks the math.

4. The Special Case: Cubic Fields (The "Three-Leaf Clover")

The paper applies this theory to a famous group of numbers called non-Galois cubic fields.

Think of these as a specific type of three-leaf clover.
The author connects these clovers to quadratic fields (two-leaf clovers) and uses a famous result from the 1930s (Davenport and Heilbronn) about the "3-leaf" patterns.
The Result: When these cubic fields are sorted by the product of their heavy stamps (ramified primes), the "3-part" of their fingerprint behaves wildly. The probability of it being small is zero, and the average size is infinite.

5. How They Did It: The "Tower" Strategy

To prove this, the author builds a mathematical "tower."

The Tower: Imagine a tower where each floor represents a layer of the group's structure. The bottom floor is a simple, abelian group (like a circle of friends holding hands). The floors above are built on top of this.
The Logic: The author uses a tool called Class Field Theory (which is like a universal translator between the structure of the group and the distribution of prime numbers).
They show that because the bottom floor of the tower is "heavy" (contains a specific prime factor), it forces the entire tower to have a specific behavior. This forces the "heavy stamps" (ramified primes) to appear in a way that creates the infinite average.

Summary of the Claim

The paper does not claim to solve real-world problems like cryptography or physics. It is a pure mathematics proof.

The core claim is:
If you look at a specific family of number fields (those with a solvable structure containing a normal abelian subgroup) and you sort them by the product of their ramified primes, you will find that:

The chance of finding a "small" class group is zero.
The average size of the class group is infinite.

This confirms a "wild" statistical behavior that was predicted by heuristics (guesses) but had not been rigorously proven for these specific types of fields until now. The author uses a combination of counting techniques and the "tower" of group structures to show that the math forces this infinite explosion.

Technical Summary: Infinite Moments of Class Groups for Solvable Fields with a Normal Abelian Subgroup

Problem Statement
The paper investigates the statistical distribution of class groups of number fields, specifically focusing on the "moments" of these groups. The central problem addressed is the phenomenon of "zero-probability" and "infinite moments" for specific $p$ -parts of class groups. While the Cohen-Lenstra-Martinet heuristics predict specific probability distributions for class groups of number fields, these heuristics often fail or require modification when the Galois group is solvable and contains a normal abelian subgroup, particularly when the prime $p$ divides the order of this subgroup.

Specifically, the paper aims to prove that under certain conditions on the ordering of number fields (using generalized discriminants) and the structure of the Galois group, the probability that the $p$ -rank of the class group is bounded by any integer $r$ is zero. Consequently, the expected value of the number of homomorphisms from the class group to $\mathbb{Z}/p\mathbb{Z}$ (the $\mathbb{Z}/p\mathbb{Z}$ -moment) is infinite. This contrasts with the behavior of other Sylow $p$ -subgroups where the Cohen-Lenstra heuristics typically predict finite moments.

Methodology
The author employs a combination of algebraic number theory, class field theory, and analytic number theory techniques. The methodology proceeds through the following steps:

Galois Algebras and Local Specifications: The paper utilizes the framework of Galois $(G, F)$ -algebras to generalize the notion of field extensions. This allows for a precise definition of "local specifications" at primes, which encode the ramification behavior of extensions. The author establishes a correspondence between isomorphism classes of Galois algebras and conjugacy classes of continuous homomorphisms from the absolute Galois group to $G$ .
Generalized Discriminants: A counting function for number fields is defined via a "generalized discriminant." This is constructed using a weight function $c_G: G \to \mathbb{R}_{\ge 0}$ that assigns values to elements of the Galois group based on their order and conjugacy class. The generalized discriminant of a field extension is the product of local contributions determined by the inertia generators of the extension at each prime.
Restricted Ramification Bounds: A key technical component is the derivation of upper bounds for the number of solutions to restricted ramification problems for solvable extensions. By decomposing a solvable group $G$ into a tower of abelian quotients, the author applies the Minkowski bound iteratively. This yields an estimate for the number of fields with a fixed local specification, bounded by a power of the discriminant (or generalized discriminant) and the size of the class group of intermediate fields.
Field Counting and Moment Analysis: The paper analyzes the asymptotic behavior of the counting function $N_{C, C}(X)$ (the number of fields with generalized discriminant less than $X$ ) and the subfamilies $N_{C, C}^\gamma(X)$ (fields with exactly $\gamma$ ramified primes satisfying specific inertia conditions). By comparing the growth rates of these counting functions, the author demonstrates that fields with a higher number of specific ramified primes dominate the distribution.
Application to Class Group Ranks: Using the established bounds on field counting, the author applies a result from previous work (Wang [22]) which links the $p$ -rank of the class group to the number of primes with specific inertia generators. If the set of fields with a high number of such primes grows sufficiently fast relative to the total set of fields, the $p$ -rank is forced to be large, leading to the zero-probability and infinite moment results.

Key Contributions and Results

Theorem 1.11 (General Solvable Case): For a transitive permutation group $G$ containing a normal abelian subgroup $N$ such that $\gcd(|N|, |G/N|) = 1$ , and for a generalized discriminant defined by a weight function where the minimum weight is achieved only by elements in $N$ of order divisible by $p$ , the paper proves:
- The probability that the $p$ -rank of the class group is $\le r$ is 0 for any non-negative integer $r$ .
- The $\mathbb{Z}/p\mathbb{Z}$ -moment of the class group is infinite.
  This result applies to fields ordered by generalized discriminants associated with the group structure.
Theorem 1.12 (Non-Galois Cubic Fields): The paper applies the general theory to non-Galois cubic fields ordered by the product of ramified primes. By relating these fields to Galois $S_3$ -extensions and utilizing the known $\mathbb{Z}/3\mathbb{Z}$ -moment results for quadratic fields (Davenport-Heilbronn) and field-counting results for cubic fields ordered by the radical of the discriminant (Shankar and Thorne), the author proves:
- For non-Galois cubic fields ordered by the product of ramified primes, the $\mathbb{Z}/3\mathbb{Z}$ -moment of the class group is infinite.
- The probability that the $3$-rank is bounded is zero.
Upper Bounds for Solvable Extensions: The paper provides explicit upper bounds for the number of solvable extensions with restricted ramification (Theorem 4.6 and Proposition 5.6). These bounds are expressed in terms of the discriminant and the structure of the Galois group tower, showing that the number of such fields grows polynomially with the discriminant.
Product of Ramified Primes: The paper extends the analysis to the case where the counting function is the product of ramified primes (Theorem 6.8 and 6.9). Under a hypothesis regarding the moments of class groups of the base fields in a tower (Hypothesis 6.5), the author proves that the infinite moment phenomenon persists even when the ordering is based on the product of ramified primes rather than the standard discriminant.

Significance and Claims
The paper claims to provide a rigorous proof of the "infinite moment" phenomenon for class groups in specific solvable settings, confirming a qualitative difference in the distribution of the $p$ -part of the class group when $p$ divides the order of a normal abelian subgroup.

Motivation: The work is motivated by the failure of the original Cohen-Lenstra heuristics to predict the behavior of the 2-part of class groups in quadratic fields (a phenomenon known as "zero-probability" for bounded rank) and the subsequent Gerth's Conjecture. The paper extends this observation to general solvable groups and other primes.
Scope: The results are conditional on the existence of suitable weight functions and, in the case of the product of ramified primes, on a hypothesis regarding the moments of class groups of intermediate fields.
Specific Application: The paper explicitly highlights the case of non-Galois cubic fields. It claims that for these fields, when ordered by the product of ramified primes, the $\mathbb{Z}/3\mathbb{Z}$ -moment is infinite. This result relies on combining the author's general framework with existing results on quadratic fields and cubic field counting.

The paper does not propose new experimental applications or future directions beyond the mathematical proofs presented. Its significance lies in unifying the understanding of class group statistics across different types of solvable extensions and ordering functions, demonstrating that the "infinite moment" behavior is a robust feature under specific group-theoretic and arithmetic conditions.