Exploring the 2-Part of Class Groups in Quadratic Fields: Perspectives on the Cohen–Lenstra Conjectures

Abstract

Cohen and Lenstra introduced conjectures concerning the distribution of class numbers in quadratic fields, though many of these conjectures remain unproven. This paper investigates the 2-part of class groups in imaginary quadratic fields and examines their alignment with the Cohen–Lenstra heuristics. We provide detailed proofs of key theorems related to ideal decompositions and modular homomorphisms, and we explore the distribution of class groups of imaginary quadratic fields. Our analysis includes constructing imaginary quadratic fields with prescribed 2-class groups and discussing the implications of these findings on the Cohen–Lenstra conjecture.

1. Introduction

Quadratic fields, denoted as $K=\mathbb{Q}\left(\sqrt{d}\right)$, where d is a non-square integer, are foundational structures in algebraic number theory. When $d>0$, K represents a real quadratic field containing real roots, while, for $d<0$, K is an imaginary quadratic field containing complex roots. The class number of a quadratic field, a critical invariant in number theory, quantifies the deviation from unique factorization within the ring of integers of K. This invariant is essential for theoretical advancements in number theory and has significant implications for cryptographic systems and algebraic geometry, given its role in analyzing factorization properties within quadratic fields. Quadratic fields and their associated class groups are fundamental for understanding class number distributions [1], a central issue in number theory.

Existing research has demonstrated systematic deviations in the distribution of the 2-part of class groups in imaginary quadratic fields, which are intricately linked to genus theory. This paper builds upon these studies by providing large-scale numerical verifications of these deviations. Focusing on higher-order Sylow subgroup distributions across broad discriminant ranges, we offer empirical support for theoretical findings by Gerth and others. Our results also document fine-grained statistical patterns that have not been systematically explored, contributing to a more comprehensive understanding of the 2-part of class groups in quadratic fields.

Imaginary quadratic fields, in particular, have been extensively studied due to their unique algebraic properties and historical significance. Gauss’s conjecture, which has been resolved for imaginary quadratic fields but remains open for real quadratic fields, highlights their importance in the study of class numbers. Specifically, the question of whether there exist infinitely many real quadratic fields with class number one is still unresolved and deeply tied to the structure of fundamental units in these fields. Moreover, the growth of the regulator in real quadratic fields further complicates computations, making the analysis of class numbers for real fields significantly more challenging than for imaginary fields [2].

Calculating class numbers for general number fields remains a challenging problem. Cohen and Lenstra proposed influential conjectures that provide a heuristic framework for understanding the statistical distribution of class numbers in number fields, particularly focusing on real and imaginary quadratic fields. Their conjectures suggest that, for a given discriminant, the class groups of quadratic fields exhibit probabilistic distributions favoring those with smaller orders. These heuristics, however, are explicitly limited in scope, excluding predictions for the 2-part of class groups in imaginary quadratic fields due to the complexities introduced by genus theory and other structural influences [3,4].

Recent advancements in the study of quadratic fields and their class groups have significantly refined Cohen and Lenstra’s seminal heuristics, enhancing our understanding of the intricate statistical patterns underlying class group distributions. The Cohen–Lenstra conjectures predict that the p-parts of class groups, particularly in imaginary quadratic fields, exhibit specific statistical behaviors. Nevertheless, empirical data reveal substantial deviations from these predictions in the 2-part of the class groups, which are likely influenced by the structural properties arising from genus theory. Gerth’s foundational work [5] provides a rigorous mathematical framework to analyze these deviations, establishing a direct relationship between the 4-class ranks and the principal genus of quadratic fields. By leveraging Rédei matrices and Legendre symbols, he derived explicit density formulas for the 2-class groups, contributing insights that not only align with, but also refine the Cohen–Lenstra heuristics to account for these observed deviations. Subsequent research by Fouvry and Klüners [6] further validated Gerth’s findings and extended their implications to broader statistical contexts, shedding light on the intricate structural properties of class groups. Modern computational methods, including those developed by Bhargava and Shankar, primarily focus on the odd part of the class number. While their methods provide a framework for studying class number distributions, our work extends these computational approaches to focus specifically on the 2-part of class groups in imaginary quadratic fields.

1.1. Our Contribution

This study provides a detailed examination of the 2-part of class groups in imaginary quadratic fields to assess their alignment with Cohen–Lenstra heuristics and to identify any systematic deviations. By combining modern computational techniques with theoretical frameworks, this paper explores structural patterns in class groups within these fields in the distribution of the 2-part. This research contributes valuable insights to both foundational studies in quadratic fields and broader applications in number theory and cryptography.

To contextualize the discussion of these conjectures, we first review several foundational concepts that are essential to understanding the relationship between class numbers and class groups in imaginary quadratic fields. For an imaginary quadratic field $K=\mathbb{Q}\left(\sqrt{-d}\right)$, the class number is closely connected to the Dedekind zeta function ${\zeta }_K\left(s\right)$ and the Minkowski bound, which provides a lower limit on the class group size. Moreover, the structure of the 2-part of the class group is intricately linked to the properties of the quadratic form class group, underscoring its importance in theoretical and computational settings.

1.2. Significance of the Current Study

The findings of this study hold substantial implications for both theoretical number theory and applied fields such as cryptography. Quadratic fields and their class groups are integral to understanding algebraic number theory, and an in-depth analysis of the 2-part distribution in these groups serves multiple important purposes.

Theoretical Impact on the Class Number Problem and Conjecture Refinement: By investigating deviations in the 2-part distribution from expected heuristic predictions, this research addresses core questions about class number distribution in quadratic fields. These insights are valuable for refining the Cohen–Lenstra conjectures, which could lead to more accurate predictive models for class group distributions. This is especially relevant for imaginary quadratic fields, where genus theory adds complexity to the underlying structure of the 2-part.

Implications for Cryptographic Algorithm Design: The structures of class groups in quadratic fields, particularly the 2-part distributions in imaginary quadratic fields, offer significant insights into cryptographic applications. These insights can be summarized as follows.

• Factorization and Discrete Logarithm Problems: The 2-part of class groups is closely linked to the structural properties of quadratic forms and genus theory, which directly influence the computational complexity of core cryptographic tasks such as integer factorization and the discrete logarithm problem. This understanding helps to evaluate the difficulty of these problems in cryptographic contexts more precisely.
• Patterns and Algorithmic Design: This study identified systematic deviations in the 2-part distributions from Cohen–Lenstra heuristics, revealing fine-grained statistical patterns, such as periodic behaviors in higher-order Sylow subgroup distributions. These findings enable the design of secure and efficient cryptographic algorithms, particularly in the context of key generation based on specific class group structures.
• Applications in Key Exchange Protocols: Leveraging tools like SageMath, the structural analysis of the 2-part facilitates the development of optimized key exchange protocols. These protocols can exploit the intricate properties of the class groups to achieve higher levels of security and computational efficiency.
• Vulnerability Assessments: Deviations from heuristic predictions provide a framework for assessing vulnerabilities in cryptographic systems. Fields with anomalous class group structures, particularly in the 2-part, can be analyzed to identify their suitability or risks for cryptographic applications.

The integration of these results into computational frameworks bridges the gap between theoretical advancements and practical cryptographic applications. This study highlights the importance of class group analyses in informing both the foundational understanding and applied design of secure cryptographic systems.

Advancements in Large-Scale Numerical Analyses in Number Theory: This study demonstrates the utility of large-scale, high-performance computational techniques for analyzing the 2-part distributions of class groups in imaginary quadratic fields. By leveraging computational frameworks such as SageMath, this research enables efficient and scalable simulations of class group structures over broad discriminant ranges. These simulations provide empirical validation for theoretical models and uncover fine-grained statistical patterns that extend our understanding of class group distributions.

The computational framework developed in this study contributes to algebraic number theory by

• Facilitating the systematic verification of heuristic models, such as those refined from the Cohen–Lenstra conjectures, through comparisons with extensive empirical data.
• Providing a scalable methodology for large-scale class group computations, which can be directly integrated into software frameworks like SageMath to support further research.

These advancements illustrate the transformative potential of modern computational methods in addressing long-standing theoretical challenges, such as identifying systematic deviations from heuristic predictions. By establishing a reproducible and robust computational approach, this study lays the groundwork for extending numerical analyses to broader contexts in algebraic number theory, including the statistical modeling of Sylow subgroup distributions and other class group properties.

1.3. Novel Contributions of This Study

The distributions of the 2-part of class groups in quadratic fields, particularly imaginary fields, have been extensively studied in the number theory community. Foundational works, including those by Gerth [5,7], Cohen–Lenstra [3], and subsequent refinements by Malle [8], Washington [9] and Bhargava et al. [10], provide critical insights into the densities and structures of Sylow subgroups.

This study builds upon these foundational works by exploring the systematic deviations observed in the 2-part of class groups of imaginary quadratic fields. Specifically, this study makes the following contributions:

• Theoretical Extension: Extending the work of Gerth and Cohen–Lenstra, this study investigates deviations in the 2-part distribution, uncovering new structural patterns influenced by genus theory.
• Numerical Insight: Leveraging advanced computational methodologies, this research examines class group data over an unprecedented range of discriminants, revealing periodic behaviors and anomalies previously unobserved.
• Methodological Innovation: By refining computational techniques, this work achieves higher precision in analyzing class group distributions, supporting the validation of heuristic models and their applicability to broader ranges.

These findings not only affirm the heuristic models, but also suggest that genus theory may influence class group structures, offering a broadened perspective on quadratic field properties.

2. Fractional Ideals and Class Groups

Definition 1 

([11,12]). Let K be a number field and ${\mathcal{O}}_K$ its ring of integers. A subset I of K is called a fractional ideal of K if there exists a non-zero element $u\in K$ such that $uI$ is a non-zero ideal of ${\mathcal{O}}_K$.

Definition 2 

([13,14]). Let ${\mathcal{O}}_K$ be a Dedekind ring. A principal fractional ideal is a fractional ideal of the form $\alpha {\mathcal{O}}_K$, which is generated by a single element α in the quotient field of ${\mathcal{O}}_K$, where $\alpha \ne 0$. The group of fractional ideals modulo the group of principal ideals (i.e., non-zero principal fractional ideals), which is called the ideal class group of ${\mathcal{O}}_K$. Denote by $P\left(K\right)$, the set of all principal fractional ideals. The principal fractional ideals form a group called the principal fractional ideal group.

Let $I\left(K\right)$ denote the set of fractional ideals of the number field K, and the quotient group $Cl\left(K\right)=I\left(K\right)/P\left(K\right)$ is called the ideal class group (or simply the class group) of K. An ideal class of K is an element of $Cl\left(K\right)$. Therefore, two fractional ideals are equivalent in K if they lie in the same coset of $I\left(K\right)/P\left(K\right)$. Both $I\left(K\right)$ and $P\left(K\right)$ are infinite abelian groups, but the quotient group $Cl\left(K\right)$ is a finite abelian group. The order of this group, $h\left(K\right)=\left|Cl\right(K\left)\right|$, is called the ideal class number (or simply the class number) of K.

It can be observed that $h\left(K\right)$ is an important invariant of K. From the definition of $Cl\left(K\right)$, we have

$$\begin{array}{cc}\hfill h\left(K\right)=1& ⟺I\left(K\right)=P\left(K\right)(\mathrm{i}.\mathrm{e}.,\mathrm{every}\mathrm{fractional}\mathrm{ideal}\mathrm{is}\mathrm{a}\mathrm{principal}\mathrm{fractional}\mathrm{ideal});\hfill \\ \hfill & ⟺\mathrm{every}\mathrm{ideal}\mathrm{in}{\mathbb{Z}}_K\mathrm{is}\mathrm{a}\mathrm{principal}\mathrm{ideal};\hfill \\ \hfill & ⟺{\mathbb{Z}}_K\mathrm{is}\mathrm{a}\mathrm{principal}\mathrm{ideal}\mathrm{domain};\hfill \\ \hfill & ⟺{\mathbb{Z}}_K\mathrm{is}\mathrm{a}\mathrm{unique}\mathrm{factorization}\mathrm{domain}.\hfill \end{array}$$

Thus, the size of the class number $h\left(K\right)$ measures the difference between the Dedekind domain ${\mathbb{O}}_K$ and a unique factorization domain.

Definition 3 

([13,14]). Let K be a field and $O_K$ be a domain. A set $\{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n\}$ is an integral basis of $O_K$ (or K) if, for every element $\alpha \in O_K$, there is a unique representation:

$$\alpha ={\lambda }_1{\alpha }_1+\cdots +{\lambda }_n{\alpha }_n,{\lambda }_i\in \mathbb{Z}.$$

Theorem 1 

(Hermite–Minkowski). Let n be a fixed positive integer, and let $B>0$. There exist only finitely many number fields K of degree n such that $\left|d\right(K\left)\right|\le B$, where $d\left(K\right)$ denotes the discriminant of K. In particular, for quadratic fields $K=\mathbb{Q}\left(\sqrt{d}\right)$, the discriminant uniquely determines K, and the Minkowski bound ensures that each ideal class in $Cl\left(K\right)$ contains an ideal of norm $\le M\left(K\right)=\sqrt{\left|d\right(K\left)\right|/\pi }$.

Remark 1. 

The Hermite theorem, first established in 1848, asserts that, for any integer discriminant d, there are only finitely many quadratic fields K such that $d\left(K\right)=d$. This result provides the foundation for studying the classification of quadratic fields. A detailed proof of Hermite’s theorem can be found in Neukirch’s Algebraic Number Theory [15] (Theorem 2.16, Chapter III, Section 2). Furthermore, the Hermite–Minkowski theorem generalizes this result to number fields of an arbitrary degree, guaranteeing the finiteness of number fields with bounded discriminants.

For quadratic fields $K=\mathbb{Q}\left(\sqrt{d}\right)$, Hermite’s theorem simplifies the classification to K, which is uniquely determined by d. The Minkowski bound $M\left(K\right)=\sqrt{\left|d\right|/\pi }$ plays a critical role in ideal class group computations as it ensures that each ideal class contains a representative ideal with norm bounded by $M\left(K\right)$. This connection between discriminants and ideal norms is essential for the computational framework of class groups.

In recent years, advancements in computational techniques have allowed for the application of Hermite’s theorem on larger datasets, affirming its utility in both theoretical and applied contexts. These developments include algorithmic approaches that extend the reach of Hermite’s finiteness results to more complex quadratic fields, reinforcing the theorem’s relevance in current number theory research [16,17].

The Minkowski Bound and Class Group Computations

Class group computations are grounded in foundational results such as Theorem 1 and Dirichlet’s class number formula [18]. The Minkowski bound $M\left(K\right)=\sqrt{\left|d\right|/\pi }$ provides a theoretical upper limit on the norms of ideal representatives in $Cl\left(K\right)$, ensuring that every ideal class contains a representative ideal with norm $\le M\left(K\right)$. This result follows directly from the Hermite–Minkowski theorem, which establishes the finiteness of number fields with bounded discriminants.

Although the Minkowski bound provides a rigorous theoretical foundation, its direct application to practical class group computations, particularly for imaginary quadratic fields, is limited by its computational inefficiency. Instead, methods such as Gauss’s enumeration of reduced quadratic forms are preferred due to their compatibility with the arithmetic structure of quadratic fields. These methods enable precise computation of class numbers and identification of ideal class representatives with reduced computational complexity.

In this study, the Minkowski bound was leveraged as a theoretical tool to underline the finiteness properties of class groups and to establish the relationship between discriminants and the structure of $Cl\left(K\right)$. Practical computations, however, were carried out using domain-specific algorithms optimized for quadratic fields, demonstrating both efficiency and precision.

Step 1:

Computing the Minkowski bound.

The Minkowski bound for a number field K is defined as

$$M\left(K\right)={\left({\displaystyle \frac{4}{\pi }}\right)}^{r_2}{\displaystyle \frac{n!}{n^n}}{\left|d\left(K\right)\right|}^{1/2},$$

where:

• $r_1$: the number of real embeddings of K;
• $r_2$: the number of pairs of complex embeddings of K;
• $n=[K:\mathbb{Q}]=r_1+2r_2$: the degree of K;
• $\left|d\right(K\left)\right|$: the absolute value of the discriminant of K.

For imaginary quadratic fields, where $r_1=0$ and $r_2=1$, the Minkowski bound simplifies to

$$M\left(K\right)={\displaystyle \frac{2}{\pi }}\sqrt{\left|d\right(K\left)\right|}.$$

Consider $K=\mathbb{Q}\left(\sqrt{-23}\right)$, where $d\left(K\right)=-23$. In this case, we compute

$$M\left(K\right)={\displaystyle \frac{2}{\pi }}\sqrt{23}\approx 3.89.$$

This result implies that only primes $p\le 3$ need to be considered. Factoring these primes within $O_K$, the ideal classes $\left[{\mathfrak{p}}_2\right]$ and $\left[{\mathfrak{p}}_3\right]$ generate $Cl\left(K\right)$. Verification through the resolution of norm equations and examination of the group closure properties confirms that $Cl\left(K\right)$ is cyclic of order 3, with $h\left(K\right)=3$.

Step 2:

Practical computation of class groups.

For imaginary quadratic fields, the practical computation of $Cl\left(K\right)$ is most effectively achieved through the enumeration of reduced quadratic forms. This approach exploits the established correspondence between ideal classes in $Cl\left(K\right)$ and the equivalence classes of binary quadratic forms of the same discriminant. Such methods not only ensure completeness of the generating set, but also rigorously validate its independence and closure properties.

Each rational prime $p\le M\left(K\right)$ is factored in $O_K$, the ring of integers of K, as

$$p{O}_K={\mathfrak{p}}_1^{a_1}{\mathfrak{p}}_2^{a_2}\cdots {\mathfrak{p}}_g^{a_g},$$

where ${\mathfrak{p}}_i$ are distinct prime ideals. By analyzing these factorizations, the ideal classes $\left[{\mathfrak{p}}_i\right]$ form a generating set for $Cl\left(K\right)$. Rigorous verification of the independence and closure properties of these generating sets ensures their completeness and correctness in representing the class group structure.

This study highlights the efficacy of reduced-form enumeration in determining the cyclicity and overall structure of $Cl\left(K\right)$, particularly for fields with small discriminants, as demonstrated in the case $K=\mathbb{Q}\left(\sqrt{-23}\right)$.

Step 3:

Validation via Dirichlet’s class number formula.

Dirichlet’s class number formula for imaginary quadratic fields relates $h\left(K\right)$, the discriminant $\left|d\right(K\left)\right|$, and the residue of the Dedekind zeta function ${\zeta }_K\left(s\right)$ at $s=1$:

$$h\left(K\right)={\displaystyle \frac{w\sqrt{\left|d\right(K\left)\right|}}{2\pi }}{\mathrm{Res}}_{s=1}{\zeta }_K\left(s\right),$$

where w is the number of roots of unity in K ($w=2$ for imaginary quadratic fields).

For $K=\mathbb{Q}\left(\sqrt{-23}\right)$, numerical computation of ${\mathrm{Res}}_{s=1}{\zeta }_K\left(s\right)$ confirms $h\left(K\right)=3$. This agreement with the theoretical predictions obtained via reduced-form enumeration demonstrates the validity of the computational approach and underscores the interplay between numerical methods and theoretical frameworks.

Step 4:

On lattice-based methods.

Lattice-based methods, such as the Lenstra–Lenstra–Lovász (LLL) algorithm [19], are widely recognized for their efficiency in solving high-dimensional lattice problems and have broad applications in computational number theory. However, for imaginary quadratic fields, the arithmetic simplicity of these fields allows for the use of specialized techniques, such as Gauss’s method of reduced quadratic form enumeration, which are more efficient and context-specific.

This study employs methods tailored to imaginary quadratic fields, leveraging their unique structural properties to achieve both computational efficiency and theoretical rigor. While lattice-based algorithms remain essential in broader contexts, their direct application to quadratic fields is limited due to the availability of more optimized approaches. By aligning the computational framework with the inherent characteristics of imaginary quadratic fields, this study highlights the importance of selecting domain-specific methods in advancing class group computations.

Definition 4. 

If $\mathfrak{p}\in \Gamma$, the norm of $\mathfrak{p}$ is defined as $N\left(\mathfrak{p}\right)=|A/\mathfrak{p}|$.

Definition 5. 

Let $G_1$ and $G_2$ be A-modules. We write $G_1\subseteq G_2$ to indicate that $G_1$ is a submodule of $G_2$.

If $\mathfrak{p}\in \Gamma$ is a prime ideal of A, and G is a finite A-module, the $\mathfrak{p}$-rank of G, denoted by $r_{\mathfrak{p}}\left(G\right)$, is defined as the dimension of the vector space $G/\mathfrak{p}G$ over the field $A/\mathfrak{p}$. Explicitly,

$$r_{\mathfrak{p}}\left(G\right)={dim}_{A/\mathfrak{p}}(G/\mathfrak{p}G).$$

Definition 6. 

Let k be a positive integer or ∞. If $k\ne \infty$ and G is a finite A-module, then $s_k\left(G\right)$ (or $s_k^A\left(G\right))$ represents the number of A-epimorphisms from $A^k$ to G. Define

$$S_k\left(G\right):=\{\phi \in {\mathrm{Hom}}_A(A^k,G):\phi \mathit{is}\mathit{an}\mathit{epimorphism}\},s_k\left(G\right)=\left|S_k\left(G\right)\right|.$$

The value $s_k\left(G\right)$ quantifies the number of surjective homomorphisms from a free module $A^k$ to G.

If G is a finite A-module, then $w_k\left(G\right)=s_k{\left(G\right)\left|G\right|}^{-k}{\left|\mathrm{Aut}\left(G\right)\right|}^{-1}$ is the k-weight, and $w\left(G\right)=w_{\infty }\left(G\right)={\left|\mathrm{Aut}\left(G\right)\right|}^{-1}$. If $\mathfrak{p}\in \Gamma$, let

$${\eta }_k\left(\mathfrak{p}\right)=\prod _{1\le i\le k}\left(1-{\left(N\mathfrak{p}\right)}^{-i}\right),{\eta }_{\infty }\left(\mathfrak{p}\right)=\prod _{i\ge 1}\left(1-{\left(N\mathfrak{p}\right)}^{-i}\right).$$

Definition 7. 

Since every finite A-module G can be written as $G={⨁}_{i}A/{\mathfrak{p}}_i^{a_i}$, define

$${\chi }_A\left(G\right)=\prod _i{\mathfrak{p}}_i^{a_i}.$$

If $A=\mathbb{Z}$, then ${\chi }_{\mathbb{Z}}\left(G\right)=n\mathbb{Z}$, where $n=\left|G\right|$. Let α be an integral ideal, and define the k-weight $w_k\left(\alpha \right)$ of α as

$$w_k\left(\alpha \right)=\sum _{G\left(\alpha \right)}{w}_k\left(G\right),w\left(\alpha \right)=w_{\infty }\left(\alpha \right),$$

where ${\sum }_{G\left(\alpha \right)}$ denotes summation over G up to A-isomorphism with ${\chi }_A\left(G\right)=\alpha$.

The concept of ${\chi }_A\left(G\right)$ provides a compact way to encode the structure of G using ideals. The k-weight $w_k\left(\alpha \right)$ measures the contribution of all modules associated with α to certain combinatorial or arithmetic quantities, such as lattice point enumeration in ideal-related spaces.

Definition 8. 

Let G be an abelian group and p a prime number. If for every $a\in G$ there exists $n\ge 1$ such that $p^{n}a=0$, then G is called a p-primary group. For a general abelian group G, let $G_p=\{a\in G:n\ge 1,p^{n}a=0\}$ denote the p-part of G.

Theorem 2. 

Every finite abelian group is a direct sum of finite cyclic groups of prime power order. More generally, every finite abelian group is a direct sum of finite cyclic groups (see [20] (Theorem 5.13)).

Definition 9. 

A module J is called projective if there exists another module M such that $F\cong J\oplus M$, where F is a free module (see Chapter III, Section 4 in [21]).

Theorem 3. 

If J is a finitely generated projective module over a principal ideal domain (PID), then J is free.

Proof. 

By definition, a module J is projective if and only if it is a direct summand of a free module, that is, there exists a free module F and a module M such that $F\cong J\oplus M$. In other words, F can be decomposed as the direct sum of J and another module M.

Since J is a finitely generated module over a principal ideal domain (PID), we apply the structure theorem for finitely generated modules over a PID (see [21] (Theorem 7.3)). The structure theorem states that every finitely generated module E over a PID can be decomposed as

$$E\cong E_{\mathrm{tor}}\oplus F,$$

where $E_{\mathrm{tor}}$ is the torsion submodule (elements annihilated by some non-zero element of the ring), and F is a free module.

For J, since it is a direct summand of a free module and free modules are torsion-free, J itself is torsion-free. This implies that the torsion part $E_{\mathrm{tor}}$ is trivial, i.e., $E_{\mathrm{tor}}=0$. Thus, J is free. □

Definition 10. 

Let S be a subset of A. Then, S is called a multiplicatively closed set in A, if S satisfies the following two conditions:

1. 

$1\in S$;

2. 

If $a,b\in S$, then $ab\in S$,

Suppose A is a domain and S is a multiplicatively closed set of A. Then, $S^{-1}A$ represents the localization of A with respect to S and is defined as

$$S^{-1}A:=\left\{{\displaystyle \frac{r}{s}}:r\in A,s\in S,{\displaystyle \frac{r}{s}}={\displaystyle \frac{r^{\prime }}{s^{\prime }}}\iff \exists u\in S\mathrm{such}\mathrm{that}u(r{s}^{\prime }-s{r}^{\prime })=0\right\}.$$

Addition and multiplication in $S^{-1}A$ are defined as follows:

$${\displaystyle \frac{a}{s}}+{\displaystyle \frac{b}{t}}={\displaystyle \frac{at+bs}{st}},\left({\displaystyle \frac{a}{s}}\right)\left({\displaystyle \frac{b}{t}}\right)={\displaystyle \frac{ab}{st}}.$$

Let $\mathfrak{p}$ be a prime ideal of A, and let $S_{\mathfrak{p}}=A\setminus \mathfrak{p}$. Then, $S_{\mathfrak{p}}^{-1}A$ is called the localization of A at $\mathfrak{p}$ and is denoted $A_{\mathfrak{p}}$. It is a local ring.

If J is a projective A-module, define the localization of J at S as $S^{-1}J$. In this case, $S^{-1}J$ is a projective module over $S^{-1}A$. The localization of a Dedekind domain is a principal ideal domain. Moreover, projective modules over a principal ideal domain are free, so projective modules over a Dedekind domain are locally free [22].

Definition 11. 

Suppose J is a finitely generated projective A-module and Γ is a set of non-zero prime ideals of A. If $\mathfrak{p}\in \Gamma$, the rank of J at $\mathfrak{p}$ is defined as the rank of $J_{\mathfrak{p}}$ as a free module over $A_{\mathfrak{p}}$, where $J_{\mathfrak{p}}$ and $A_{\mathfrak{p}}$ are the localizations of J and A at $\mathfrak{p}$. In general, the rank is a local function on Γ, but for Dedekind domains, the rank is constant.

Theorem 4 

([22]). If A is a Dedekind domain and J is a projective module, then $J\cong A\oplus I$, where I is a non-zero ideal and $\mathrm{rank}\left(J\right)=n+1$.

Note. This theorem provides a general method for determining the rank of projective modules over Dedekind domains.

3. Ideal Decompositions and Modular Homomorphisms

In this section, we delve into the foundational aspects of ideal decompositions and modular homomorphisms, which are essential for understanding the structure of class groups and their automorphisms. We provide detailed proofs of key theorems, following the exposition in [3].

Let $A={\mathcal{O}}_K$ denote the ring of integers of a number field K, and let $\Gamma$ represent the set of non-zero prime ideals of A.

Main Theorem and Proof

Theorem 5. 

Suppose J is a projective A-module with rank k and G is a finite A-module with ${\chi }_A\left(G\right)=\alpha$, then

(i) The number of A-module epimorphisms from J to G is equal to $s_k\left(G\right)$;

(ii) $s_k\left(G\right)={\left(N\alpha \right)}^k{\prod }_{\mathfrak{p}|\alpha }\left({\displaystyle \frac{{\eta }_k\left(\mathfrak{p}\right)}{{\eta }_{k-r_{\mathfrak{p}}\left(G\right)}\left(\mathfrak{p}\right)}}\right)$ and $w_k\left(G\right)={\prod }_{\mathfrak{p}|\alpha }{\displaystyle \frac{{\eta }_k\left(\mathfrak{p}\right)}{{\eta }_{k-r_{\mathfrak{p}}\left(G\right)}\left(\mathfrak{p}\right)}}·{\displaystyle \frac{1}{\left|\mathrm{Aut}\right(G\left)\right|}}$;

(iii) $\#\{H\le J:J/H\cong G\}={\left(N\alpha \right)}^k{w}_k\left(G\right)$;

(iv) ${lim}_{k\to +\infty }{w}_k\left(G\right)=w\left(G\right)$.

Proof. 

(i) Let $S_{\alpha }$ be the set of prime ideals in A excluding all prime ideals that are not divisible by $\alpha$. Define $S_{\alpha }^{-1}A$, $S_{\alpha }^{-1}J$, and $S_{\alpha }^{-1}G$ as the localization of A, J, and G, respectively. For convenience, denote $A_{\alpha }$, $J_{\alpha }$, and $G_{\alpha }$ as $S_{\alpha }^{-1}A$, $S_{\alpha }^{-1}J$, and $S_{\alpha }^{-1}G$, respectively.

At this time, $A_{\alpha }$ is a semi-local Dedekind domain, and a semi-local Dedekind domain is a principal ideal domain. Therefore, $J_{\alpha }$ as an $A_{\alpha }$-module is a free module, so we have $J_{\alpha }\cong A_{\alpha }^k$, and there exists a module isomorphism $\psi :J_{\alpha }\to A_{\alpha }^k$.

Thus, any $A_{\alpha }$-module surjection from $J_{\alpha }$ to $G_{\alpha }$ can be transformed into a surjection from $A_{\alpha }^k$ to $G_{\alpha }$ via this isomorphism. Conversely, any $A_{\alpha }$-module surjection from $A_{\alpha }^k$ to $G_{\alpha }$ can be transformed into a surjection from $J_{\alpha }$ to $G_{\alpha }$ via the isomorphism. Therefore, the number of $A_{\alpha }$-module epimorphisms from $J_{\alpha }$ to $G_{\alpha }$ is equal to $s_k\left(G_{\alpha }\right)$. □

To prove (i) in general, the following concepts and theorems are needed.

Definition 12. 

Localization of mapping: Let $\phi :M\to N$ be an A-module homomorphism. Then, the localization of φ at α is defined as

$${\phi }_{\alpha }:S_{\alpha }^{-1}M\to S_{\alpha }^{-1}N,m/u\mapsto \phi \left(m\right)/u,u\in S_{\alpha }.$$

Proposition 1. 

Suppose $\psi :A\to A_{\alpha }$ is the natural localization mapping. Then, it has the following properties:

(i) 

For any ideal $I\subseteq A_{\alpha }$, it holds that

$$I={\psi }^{-1}\left(I\right)A_{\alpha },$$

and the mapping $I\mapsto {\psi }^{-1}\left(I\right)$ is an injection from the set of ideals of $A_{\alpha }$ to the set of ideals of A, which maps prime ideals to prime ideals.

(ii) 

Suppose N is an ideal of A. Then, N has the form ${\psi }^{-1}\left(I\right)$, where $I\subseteq A_{\alpha }$, if and only if

$$N={\psi }^{-1}\left(N{A}_{\alpha }\right),$$

that is, if $a\in A$ and $au\in N$ for some $u\in A$, then $a\in N$. This correspondence $I\mapsto {\psi }^{-1}\left(I\right)$ is an isomorphism from the prime ideals of $A_{\alpha }$ to the prime ideals of A that are not contained in α. A similar result holds for any module and its submodules.

Proof. 

For the proof, see [11] (pp. 61–63). □

This property indicates the existence of a natural mapping between a ring and its localization, which establishes a correspondence between the ideals in the ring and ideals in the local ring. This facilitates the examination of ideals and prime ideals in the local ring following localization. Moreover, for Dedekind domains, where prime ideals coincide with maximal ideals, one only needs to consider the unique maximal ideal in the local ring, thus establishing a corresponding relationship between the local ring and its original counterpart.

Theorem 6. 

If $\phi :M\to N$ is an A-module isomorphism, then φ is injective, surjective, or bijective if and only if for every maximal ideal α of A, the localized mapping ${\phi }_{\alpha }:S_{\alpha }^{-1}M\to S_{\alpha }^{-1}N$ is injective, surjective, or bijective, respectively.

Proof. 

For the proof of the theorem, see [11] (pp. 67–68). □

By applying Theorem 6 and Proposition 1, one can prove Theorem 5 (i) by replacing M with $A^k$ and N with G.

Lemma 1. 

If $\phi \in {\mathrm{Hom}}_A(A^k,G)$, let $\overline{\phi }:{(A/\mathfrak{p})}^k\to G/\mathfrak{p}G$ be defined as $\overline{\phi }\left(\overline{g}\right)=\overline{\phi \left(g\right)}$, where $g\in A^k$ and $\overline{g}\in {(A/\mathfrak{p})}^k$. Then, φ is surjective if and only if $\overline{\phi }$ is surjective.

Proof. 

First, we prove that the definition of $\overline{\phi }$ is reasonable. Suppose $\overline{g_1}=\overline{g_2}$, so $\overline{g_2-g_1}=0$, and we have

$$\overline{\phi }\left(\overline{g_2}\right)-\overline{\phi }\left(\overline{g_1}\right)=\overline{\phi \left(g_2\right)}-\overline{\phi \left(g_1\right)}=\overline{\phi (g_2-g_1)}=\overline{0}.$$

It is obvious that $\overline{\phi }$ is an $A/\mathfrak{p}$-module homomorphism.

Now, since G is a p-group, we can express $G={\oplus }_{i}A/{\mathfrak{p}}^{a_i}$ and $\mathfrak{p}G={\oplus }_i\mathfrak{p}A/{\mathfrak{p}}^{a_i}$. Thus, for any $\phi \in {\mathrm{Hom}}_A(A^k,G)$, we have

$$\phi \cong {\oplus }_i{\mathrm{Hom}}_A(A^k,A/{\mathfrak{p}}^{a_i}).$$

Therefore, we can write $\phi =({\phi }_1,{\phi }_2,\dots ,{\phi }_t)$ and $\overline{\phi }=(\overline{{\phi }_1},\overline{{\phi }_2},\dots ,\overline{{\phi }_t})$, where each ${\phi }_i=\pi \circ \phi$. It follows that $\overline{\phi }$ is surjective if and only if each $\overline{{\phi }_i}$ is surjective. □

Theorem 7. 

The equality $s_k^A\left(G\right)=s_k^{A/\mathfrak{p}}(G/\mathfrak{p}G)·\#\{\phi \in {\mathrm{Hom}}_A(A^k,G):\overline{\phi }=0\}$ holds.

Proof. 

Define

$$\Phi :S_k\left(G\right)⟶S_k^{A/\mathfrak{p}}(G/\mathfrak{p}G),\phi ⟶\overline{\phi }.$$

It is clear that $\Phi$ is surjective. By the fundamental theorem of homomorphisms, we have

$$S_k\left(G\right)/\mathrm{Ker}\Phi \cong S_k^{A/\mathfrak{p}}(G/\mathfrak{p}G),$$

where $\mathrm{Ker}\Phi =\{\phi \in {\mathrm{Hom}}_A(A^k,G):\overline{\phi }=0\}$.

Thus, we conclude that

$$s_k^A\left(G\right)=s_k^{A/\mathfrak{p}}(G/\mathfrak{p}G)·\#\{\phi \in {\mathrm{Hom}}_A(A^k,G):\overline{\phi }=0\}.$$

This proves the theorem. □

Choose a set of basis $\{e_1,e_2,\cdots ,e_k\}$ for $A^k$. Since $\overline{\phi }=0⟺\mathrm{Im}\phi \subset \mathfrak{p}G⟺\phi \left(e_i\right)\in \mathfrak{p}G$ for every i and each $e_i$, the number of $\phi$ is given by $\left|\mathfrak{p}G\right|$. Therefore,

$$\#\{\phi \in {\mathrm{Hom}}_A(A^k,G):\overline{\phi }=0\}={\left|\mathfrak{p}G\right|}^k={\displaystyle \frac{{\left|G\right|}^k}{{|G/\mathfrak{p}G|}^k}}.$$

Let $r=r_{\mathfrak{p}}\left(G\right)$, then $G/\mathfrak{p}G$ is a vector space over $A/\mathfrak{p}$ of dimension r. Thus, $G/\mathfrak{p}G\cong {(A/\mathfrak{p})}^r$, and, consequently, $|G/\mathfrak{p}G|={\left(N\mathfrak{p}\right)}^r$. Therefore,

$${\left|\mathfrak{p}G\right|}^k={\displaystyle \frac{{\left|G\right|}^k}{{\left(N\mathfrak{p}\right)}^{kr}}}={\displaystyle \frac{{\left(N\alpha \right)}^k}{{\left(N\mathfrak{p}\right)}^{kr}}}.$$

On the other hand, $s_k^{A/\mathfrak{p}}(G/\mathfrak{p}G)$ represents the number of $k\times r$ matrices with rank r over $A/\mathfrak{p}$. This is equivalent to counting the number of linearly independent r-dimensional vectors $(v_1,v_2,\dots ,v_r)$ in ${(A/\mathfrak{p})}^k$.

Since a vector space of dimension i has ${\left(N\mathfrak{p}\right)}^i$ elements over $A/\mathfrak{p}$, it follows that

$$s_k^{A/\mathfrak{p}}(G/\mathfrak{p}G)=({\left(N\mathfrak{p}\right)}^k-1)(N{\mathfrak{p}}^k-N\mathfrak{p})\cdots ({\left(N\mathfrak{p}\right)}^k-{\left(N\mathfrak{p}\right)}^{r-1})={\displaystyle \frac{{\left(N\mathfrak{p}\right)}^{kr}{\eta }_k\left(\mathfrak{p}\right)}{{\eta }_{k-r}\left(\mathfrak{p}\right)}}.$$

Hence, Theorem 5 (ii) is established.

Proof of Theorem 5 (iii). 

Let $Y=\{H\subseteq J:J/H\cong G\}$ and $X=\{\mathrm{Ker}\phi :\phi \in {\mathrm{Hom}}_A(J,G),\phi \mathrm{is}\mathrm{surjective}\}$. We assert that $Y=X$.

• Clearly, $X\subseteq Y$. To complete the proof, we need only show that $Y\subseteq X$.
• Suppose $H\in Y$. Then, there exists an A-module isomorphism ${\phi }_0:J/H\to G$. Combining this with the natural projection $\pi :J\to J/H$, we obtain $\phi ={\phi }_0\circ \pi \in {\mathrm{Hom}}_A(J,G)$, which is surjective, and

  $$\mathrm{Ker}\phi =\mathrm{Ker}\pi =H.$$

Thus, $Y\subseteq X$, proving that $Y=X$. Therefore, we have

$$\#\{H\le J:J/H\cong G\}=\#\{\mathrm{Ker}\phi :\phi \in {\mathrm{Hom}}_A(J,G),\phi \mathrm{is}\mathrm{surjective}\}.$$

Since $\mathrm{Ker}{\phi }_1=\mathrm{Ker}{\phi }_2⟺\exists \sigma \in \mathrm{Aut}\left(J\right)$ such that ${\phi }_2=\sigma \circ {\phi }_1$, we deduce that

$$\#\{\mathrm{Ker}\phi :\phi \in {\mathrm{Hom}}_A(J,G),\phi \mathrm{is}\mathrm{surjective}\}={\displaystyle \frac{s_k\left(G\right)}{\left|\mathrm{Aut}\right(G\left)\right|}}=w_k\left(G\right)·{\left|G\right|}^{-k}={\left(N\alpha \right)}^k{w}_k\left(G\right).$$

Thus, Theorem 5 (iii) is proved. □

Proof of Theorem 5(iv). 

From Theorem 5(ii), taking the limit as $k\to +\infty$, we obtain

$$\underset{k\to +\infty }{lim}{w}_k\left(G\right)={\displaystyle \frac{1}{\left|\mathrm{Aut}\right(G\left)\right|}}\underset{k\to +\infty }{lim}\prod _{\mathfrak{p}\mid \alpha }{\displaystyle \frac{{\eta }_k\left(\mathfrak{p}\right)}{{\eta }_{k-r_{\mathfrak{p}}\left(G\right)}\left(\mathfrak{p}\right)}}={\displaystyle \frac{1}{\left|\mathrm{Aut}\right(G\left)\right|}}.$$

Thus, Theorem 5 (iv) holds. □

Lemma 2. 

If ${\phi }_1\in {\mathrm{Hom}}_A(A^{k_1},G_1)$ is surjective, then

$$\#\{\phi \in {\mathrm{Hom}}_A(A^{k_1+k_2},G):\phi \mathit{is}\mathit{surjective}\mathit{and}\phi {|}_{A^{k_1}}={\phi }_1\}=s_{k_2}(G/G_1){\left|G_1\right|}^{k_2}.$$

For a proof, see [3] (Lemma 3.3).

Theorem 8. 

When $k_1,k_2\ne \infty$ and G is a finite A-module, we have

$$s_{k_1+k_2}\left(G\right)=\sum _{G_1\subseteq G}{s}_{k_1}\left(G_1\right)s_{k_2}(G/G_1){\left|G_1\right|}^{k_2}.$$

Proof. 

Suppose $A^{k_1\times k_2}=A^{k_1}\times A^{k_2}$. For a given $a=(a_1,a_2)\in A^{k_1}\times A^{k_2}$ and $\phi \in {\mathrm{Hom}}_A(A^{k_1\times k_2},G)$, define

$${\phi }_1:A^{k_1}\to G,{\phi }_1\left(a\right)=\phi (a_1,0);$$

$${\phi }_2:A^{k_2}\to G,{\phi }_2\left(a\right)=\phi (0,a_2).$$

Thus, $\phi (a_1,a_2)=\phi (a_1,0)+\phi (0,a_2)={\phi }_1\left(a_1\right)+{\phi }_2\left(a_2\right)$, and we conclude that

$$s_{k_1+k_2}\left(G\right)=\sum _{G_1\subseteq G}\#\{\phi \in {\mathrm{Hom}}_A(A^{k_1+k_2},G):\phi \mathrm{is}\mathrm{surjective}\mathrm{and}\phi \left(A^{k_1}\right)=G_1\}.$$

Thus, we have

$$s_{k_1+k_2}\left(G\right)=\sum _{G_1\subseteq G}\sum _{{\phi }_1\in S_{k_1}\left(G\right)}\#\{\phi \in {\mathrm{Hom}}_A(A^{k_1+k_2},G):\phi \mathrm{is}\mathrm{surjective}\mathrm{and}\phi {|}_{A^{k_1}}={\phi }_1\}.$$

□

Theorem 9. 

Let α be a non-zero ideal of A. For any $k_2\ne \infty$, we have

$$w_{k_1+k_2}=\sum _{\beta \mid \alpha }{\left(N\beta \right)}^{-k_2}{w}_{k_1}\left(\beta \right)w_{k_2}\left(\alpha {\beta }^{-1}\right).$$

For a proof, see [3] (Theorem 3.6).

Theorem 10. 

Let α be a non-zero ideal of A. For any k, we have

$$\sum _{\beta \mid \alpha }{w}_k\left(\beta \right)=\left(N\alpha \right)w_{k+1}\left(\alpha \right).$$

In particular, ${\sum }_{\beta \mid \alpha }w\left(\beta \right)=N\left(\alpha \right)w\left(\alpha \right)$.

Proof. 

Note that $s_1\left(G\right)\ne 0$ if and only if $G\cong A/\alpha$, where $\alpha$ is a non-zero ideal of A. According to the fundamental theorem of modular homomorphisms, we have $G\cong A/\mathrm{Ker}\phi$, where $\phi \in s_1\left(G\right)$ and $s_1(A/\alpha )\cong \mathrm{Aut}(A/\alpha )$. Therefore, $s_1(A/\alpha )=\left|\mathrm{Aut}(A/\alpha )\right|$. By Theorem 3.5, setting $k_1=k$ and $k_2=1$, we obtain

$$\begin{array}{cc}\hfill w_1\left(\alpha \right)& =\sum _{G\left(\alpha \right)}{\displaystyle \frac{s_1\left(G\right)}{\left|G\right|·\left|\mathrm{Aut}\right(G\left)\right|}}={\left|G\right|}^{-1}={\displaystyle \frac{1}{N\alpha }},\hfill \\ \hfill w_{k+1}\left(\alpha \right)& =\sum _{\beta \mid \alpha }{\left(N\beta \right)}^{-1}{w}_k\left(\beta \right)w_1\left(\alpha {\beta }^{-1}\right)\hfill \\ \hfill & =\sum _{\beta \mid \alpha }{\left(N\beta \right)}^{-1}{w}_k\left(\beta \right){\left(N\left(\alpha {\beta }^{-1}\right)\right)}^{-1}\hfill \\ \hfill & =\sum _{\beta \mid \alpha }{\left(N\beta \right)}^{-1}{w}_k\left(\beta \right)N{\left(\alpha \right)}^{-1}N\left(\beta \right).\hfill \end{array}$$

Thus, we conclude that ${\sum }_{\beta \mid \alpha }{w}_k\left(\beta \right)=\left(N\alpha \right)w_{k+1}\left(\alpha \right)$. Taking the limit as $k\to \infty$, we find ${\sum }_{\beta \mid \alpha }w\left(\beta \right)=N\left(\alpha \right)w\left(\alpha \right)$. □

Theorem 11. 

Let $\mathfrak{p}\in \Gamma$ be a prime ideal.

(i) 

When $\mathrm{Re}\left(s\right)>-1$, we have

$$\sum _{i\ge 0}{w}_k\left({\mathfrak{p}}^i\right)N{\left(\mathfrak{p}\right)}^{-is}=\prod _{1\le j\le k}{\left(1-{\left(N\mathfrak{p}\right)}^{-j-s}\right)}^{-1}.$$

(ii) 

If ${\zeta }_{k,A}\left(s\right)={\zeta }_k\left(s\right)={\sum }_{\alpha }{w}_k\left(\alpha \right){\left(N\alpha \right)}^{-s}$, where $\mathrm{Re}\left(s\right)>0$, then

$${\zeta }_k\left(s\right)=\prod _{1\le j\le k}{\zeta }_A(s+j),$$

where ${\zeta }_A\left(s\right)$ is the Dedekind zeta function of A.

Proof. 

(i) For $k=1$, since $w_1\left(\alpha \right)=1/N\alpha$ and $N\left({\mathfrak{p}}^m\right)={\left(N\mathfrak{p}\right)}^m$, we have

$$\begin{array}{cc}\hfill \sum _{i\ge 0}{w}_1\left({\mathfrak{p}}^i\right){\left(N\mathfrak{p}\right)}^{-is}& =\sum _{i\ge 0}{\left(N\mathfrak{p}\right)}^{-i}{\left(N\mathfrak{p}\right)}^{-is}\hfill \\ \hfill & =\sum _{i\ge 0}{\left(N\mathfrak{p}\right)}^{-i(1+s)}\hfill \\ \hfill & ={(1-{\left(N\mathfrak{p}\right)}^{-1-s})}^{-1}.\hfill \end{array}$$

For $k\ge 2$, using Theorems 3.5 and 3.6 and the fact that $N\left({\mathfrak{p}}^m\right)={\left(N\mathfrak{p}\right)}^m$, we obtain

$$\begin{array}{cc}\hfill \sum _{i\ge 0}{w}_k\left({\mathfrak{p}}^i\right){\left(N\mathfrak{p}\right)}^{-is}& =\sum _{i\ge 0}\sum _{l\le i}{w}_{k-1}\left({\mathfrak{p}}^l\right){\left(N\mathfrak{p}\right)}^{-i}{\left(N\mathfrak{p}\right)}^{-is}\hfill \\ \hfill & =\sum _{i\ge 0}{\left(N\mathfrak{p}\right)}^{-i(1+s)}\sum _{l\le i}{w}_{k-1}\left({\mathfrak{p}}^l\right)\hfill \\ \hfill & =\sum _{l\ge 0}\sum _{i\ge l}{\left(N\mathfrak{p}\right)}^{-i(1+s)}{w}_{k-1}\left({\mathfrak{p}}^l\right)\hfill \\ \hfill & =\sum _{l\ge 0}{w}_{k-1}\left({\mathfrak{p}}^l\right){\left(N\mathfrak{p}\right)}^{-l(s+1)}{\left(1-{\left(N\mathfrak{p}\right)}^{-(1+s)}\right)}^{-1}.★\hfill \end{array}$$

Continuing this process, we have

$$\begin{array}{cc}\hfill ★& =\sum _{l\ge 0}{w}_1\left({\mathfrak{p}}^l\right){\left(N\mathfrak{p}\right)}^{-l(s+k)}\prod _{1\le j\le k}{\left(1-{\left(N\mathfrak{p}\right)}^{-(j+s)}\right)}^{-1}\hfill \\ \hfill & =\prod _{1\le j\le k}{\left(1-{\left(N\mathfrak{p}\right)}^{-(j+s)}\right)}^{-1}·\sum _{l\ge 0}{\left(N\mathfrak{p}\right)}^{-l(s+k+1)}\hfill \\ \hfill & =\prod _{1\le j\le k}{\left(1-{\left(N\mathfrak{p}\right)}^{-(j+s)}\right)}^{-1}·{(1-{\left(N\mathfrak{p}\right)}^{-(s+k+1)})}^{-1}\hfill \\ \hfill & =\prod _{1\le j\le k}{\left(1-{\left(N\mathfrak{p}\right)}^{-j-s}\right)}^{-1}.\hfill \end{array}$$

(ii) For $k=1$, we have

$${\zeta }_1\left(s\right)=\sum _{\alpha }{w}_1\left(\alpha \right){\left(N\alpha \right)}^{-s}=\sum _{\alpha }{\left(N\alpha \right)}^{-(1+s)}={\zeta }_A(s+1).$$

For $k\ge 2$, we calculate

$$\begin{array}{cc}\hfill {\zeta }_k\left(s\right)& =\sum _{\alpha }{w}_k\left(\alpha \right){\left(N\alpha \right)}^{-s}=\sum _{\alpha }\sum _{\beta \mid \alpha }{w}_{k-1}\left(\beta \right){\left(N\alpha \right)}^{-(1+s)}\hfill \\ \hfill & =\sum _{\beta }\sum _{\gamma }{w}_{k-1}\left(\beta \right){\left(N\beta \right)}^{-(1+s)}{\left(N\gamma \right)}^{-(1+s)},(\alpha =\beta \gamma )\hfill \\ \hfill & =\sum _{\beta }{w}_{k-1}\left(\beta \right){\left(N\beta \right)}^{-(1+s)}{\zeta }_A(s+1)\hfill \\ \hfill & ={\zeta }_A(s+1)\sum _{\beta }{w}_{k-2}\left(\beta \right){\left(N\beta \right)}^{-(2+s)}{\zeta }_A(s+2),\dots \hfill \\ \hfill & ={\zeta }_A(s+1){\zeta }_A(s+2)\cdots {\zeta }_A(s+k-1)\sum _{\beta }{w}_1\left(\beta \right){\left(N\beta \right)}^{-(k+s-1)}.\hfill \end{array}$$

Since $w_1\left(\beta \right)={\left(N\beta \right)}^{-1}$, we conclude that

$${\zeta }_k\left(s\right)=\prod _{1\le j\le k}{\zeta }_A(s+j).$$

Thus, the theorem is proved. □

4. On the 2-Part of Class Groups in Imaginary Quadratic Fields and Connections to the Cohen–Lenstra Conjecture

In this chapter, we compute the 2-part of the class group in imaginary quadratic fields and compare the results with the Cohen–Lenstra conjecture. From these calculations, we derive new conjectures. Before presenting these conjectures, we introduce some foundational concepts and properties to aid understanding.

We begin with the concept of partitions. For any natural number, there exists a corresponding partition so that each natural number can be expressed as a sum of partitions. For example,

$$6=6+0=5+1=4+2=4+1+1=\cdots .$$

Thus, a partition can represent a natural number $n=\left(n_i\right),n_1\ge n_2\ge \cdots \ge n_k>0$.

Let $\Omega$ represent the set of partitions of natural numbers and define ${\mathcal{G}}_p$ as the set of all finite abelian p-groups (up to isomorphism). For any finite abelian p-group, it can be expressed as ${⨁}_i{(\mathbb{Z}/p^{e_i})}^{r_i}$, where $1\le i\le k$, $k>0$, $e_1>e_2>\cdots >e_k>0$, and $r_i>0$. There is a natural isomorphism between these two sets: ${\mathcal{G}}_p\cong \Omega$.

4.1. Theorem and Conjectures

Theorem 12. 

Let $G={⨁}_i{(\mathbb{Z}/p^{e_i})}^{r_i}$, $(1\le i\le k)$, where $k>0$, $e_1>e_2>\cdots >e_k>0$, and $r_i>0$. Then, the order of the automorphism group $\mathrm{Aut}\left(G\right)$ is given by

$$\left|\mathrm{Aut}\left(G\right)\right|=\left(\prod _{1\le i\le k}\prod _{1\le s\le r_i}(1-p^{-s})\right)\prod _{1\le i,j\le k}{p}^{min(e_i,e_j)r_i{r}_j}.$$

In particular, if $H=\mathrm{Aut}\left({(\mathbb{Z}/p^e)}^r\right)$, then $\left|H\right|=p^{r^{2}e}{\prod }_{1\le s\le r}(1-p^{-s})$.

Proof. 

For a detailed proof, see [4] (Theorem 2.1). □

Conjecture 1 

(Cohen–Lenstra). Suppose p is an odd prime, and let $D^{\pm }\left(X\right)$ denote the number of real or imaginary quadratic fields whose absolute discriminant is less than X. Let G be a finite abelian p-group. Then,

$${\lambda }^{\pm }\left(G\right)=\underset{X\to \infty }{lim}{\displaystyle \frac{\left|\right\{K\in D^{\pm }\left(X\right):C{l}_p\left(K\right)\cong G\left\}\right|}{|D^{\pm }\left(X\right)|}}$$

exists, and ${\lambda }^{+}\left(G\right)=c^{+}{\left|\mathrm{Aut}\left(G\right)\right|}^{-1}{\left|G\right|}^{-1}$, while ${\lambda }^{-}\left(G\right)=c^{-}{\left|\mathrm{Aut}\left(G\right)\right|}^{-1}$, where $c^{+}$ and $c^{-}$ are constants that are independent of G.

An instance of the Cohen–Lenstra conjecture posits that nearly all cyclic groups (97.7575%) form the odd part of the class groups of imaginary quadratic fields. Though this conjecture remains unproven, it offers significant insights. Notably, Cohen and Lenstra did not make a conjecture about the 2-part of the class group as Gauss’s genus theory suggests non-randomness. However, later work indicated that the Cohen–Lenstra conjecture’s principle of inverse proportions to automorphism group orders might still apply to higher ranks like 4-rank and 8-rank. To further explore the 2-part of class groups in quadratic fields, we introduce additional concepts.

4.2. Directed Graphs and the 2-Rank of Class Groups

Definition 13. 

Let $G=(V,E)$ be a directed graph, where $V=V_1\cup V_2$ is a partition of V. The partition is odd if there exists $v_1\in V_1$ such that the number of arcs from $v_1$ to vertices in $V_2$ is odd or there exists $v_2\in V_2$ such that the number of arcs from $v_2$ to vertices in $V_1$ is odd. Otherwise, the partition is even. A graph G is said to be odd if every non-trivial partition of V is odd.

Let $K=\mathbb{Q}\left(\sqrt{-D}\right)$, where $D\ge 2$ is an imaginary quadratic field, and let $r_2$ be the 2-rank of the class group $Cl\left(K\right)$. According to Gauss’s genus theory, $r_2=t-1$, where t is the number of distinct prime factors of D. Define the directed graph $G\left(D\right)$, where the vertices are the prime factors of D, and there exists an arc $\overrightarrow{p_i{p}_j}$ if $\left({\displaystyle \frac{p_j}{p_i}}\right)=-1$, where $\left({\displaystyle \frac{p_j}{p_i}}\right)$ is the Legendre symbol.

Definition 14. 

Let $M\left(G\right)=\mathrm{diag}(d_1,\dots ,d_m)-A\left(G\right)$, where $d_{ij}={\sum }_{j=1}^m{a}_{ij}$ and $A\left(G\right)$ is the adjacency matrix. Define $r={\mathrm{rank}}_{{\mathbb{F}}_2}\left(M\left(G\right)\right)$.

This study revealed that the parity of the directed graph $G\left(D\right)$, as determined by the rank of its Laplacian matrix, significantly constrains the distribution of the class group $C{l}_2\left(K\right)$. Specifically, when the discriminant D contains an even number of prime factors, $G\left(D\right)$ is more likely to produce an odd graph structure (higher-rank Laplacian matrix). This tendency restricts the generation of complex 2-Sylow subgroups, such as ${(\mathbb{Z}/2\mathbb{Z})}^k$, where $k>n-1$. This finding highlights how the Cohen–Lenstra conjecture overlooks the impact of graph structure on distribution patterns. The data suggest that modular congruences, such as $p_i\equiv p_j\mathrm{mod}4$, strongly influence the edge formation in directed graphs. This impact is transmitted through the nested structure of the graph to larger discriminants D, forming a cumulative distribution model. This hereditary effect demonstrates that preferences in subgroup formation for smaller discriminants persist into larger ones, contrasting with the independence assumed in the Cohen–Lenstra conjecture.

Lemma 3 

([23]). The graph G is odd if and only if $r=m-1$.

Theorem 13 

([23]). Let $K=\mathbb{Q}\left(\sqrt{-D}\right)$ with $D\ge 2$, and let t be the number of distinct prime factors of D. Then, $2^{t-1}\left|\right|h_K$ if and only if the directed graph $G\left(D\right)$ is odd.

Proposition 2. 

There exists an imaginary quadratic field with an arbitrarily large absolute discriminant such that the 2-part of its class group is a 2-Sylow subgroup of order 16.

Proof. 

According to Theorem 14, we know that if the directed graph $G\left(D\right)$ is odd for $t=5$, we can obtain a 2-Sylow subgroup of order 16.

Let $K=\mathbb{Q}\left(\sqrt{-D}\right)$, with $p_1=3$, $p_2=5$, $p_3=7$, $p_4=11$, and $p_5=p$. Then, $D=3\times 5\times 7\times 11\times p$, and $-D\equiv 1\left(\mathrm{mod}4\right)$. The matrix is given by

$$M\left(G\right)=\left(\begin{array}{ccccc}0& 0& 0& 0& a_{15}\\ 0& 0& 1& 0& a_{25}\\ 1& 1& 0& 0& a_{35}\\ 0& 0& 1& 0& a_{45}\\ a_{51}& a_{52}& a_{53}& a_{54}& 0\end{array}\right).$$

According to Lemma 2, to make the graph $G\left(D\right)$ odd, we need the rank of the matrix $M\left(G\right)$ to be 4. Take a special case: let $a_{15}=0,a_{25}=0,a_{51}=0,a_{52}=0,a_{53}=1,a_{54}=1$, that is, $\left({\displaystyle \frac{3}{p}}\right)=1$, $\left({\displaystyle \frac{5}{p}}\right)=1$, $\left({\displaystyle \frac{p}{3}}\right)=1$, $\left({\displaystyle \frac{p}{5}}\right)=1$, $\left({\displaystyle \frac{7}{p}}\right)=-1$, and $\left({\displaystyle \frac{11}{p}}\right)=-1$.

For the congruence equation $\left({\displaystyle \frac{3}{p}}\right)=1$ and $\left({\displaystyle \frac{p}{3}}\right)=1$, we obtain the solution $p\equiv 1\left(\mathrm{mod}12\right)$. For $\left({\displaystyle \frac{5}{p}}\right)=1$ and $\left({\displaystyle \frac{p}{5}}\right)=1$, we obtain $p\equiv 1,49\left(\mathrm{mod}60\right)$. Taking $p\equiv 1\left(\mathrm{mod}60\right)$ and combining this with $\left({\displaystyle \frac{7}{p}}\right)=-1$ and $\left({\displaystyle \frac{p}{7}}\right)=-1$, we obtain $p\equiv 61,481,901\left(\mathrm{mod}4620\right)$. At this time, $-D=3\times 5\times 7\times 11\equiv 1\left(\mathrm{mod}4\right)$. According to the prime number theorem in Dirichlet’s arithmetic progression, there are infinitely many such prime numbers. Thus, the proposition is proved. □

By Proposition 2, one can construct an infinite number of imaginary quadratic fields, where the 2-part of the class group forms a 2-Sylow subgroup of order 16. Similarly, there exist infinitely many 2-Sylow subgroups of order 8 that can be constructed. In accordance with the principles of the Cohen–Lenstra conjecture, investigations into the 2-part of class groups can be conducted to explore whether they exhibit behavior analogous to the conjecture’s predictions. Numerical calculations were performed separately for real and imaginary quadratic fields, focusing on the orders of their respective 4th-, 8th-, 16th-, and 32nd-order Sylow subgroups.

4.3. Computational Tools and Environment

This study conducted numerical simulations on a large set of imaginary quadratic fields $K=\mathbb{Q}\left(\sqrt{-d}\right)$, where the discriminant d is chosen within the range of 1 to ${10}^8$. The numerical simulations were performed on a high-performance computing platform using the following tools.

• SageMath: SageMath was utilized to compute the specific structure of the 2-part of class groups. Its extensive number theory libraries facilitate efficient computations for ideal decomposition and class group calculations.
• Class Group Computation: The 2-part of the class group is the core of this study, and the computational process includes
  • 2-Part Extraction: For each imaginary quadratic field, the class group is computed and elements related to the 2-part of the class group (i.e., elements of order a power of 2) are extracted.
  • Statistical Deviation Analysis: The computed results are compared with statistical predictions from the Cohen–Lenstra heuristics to assess conformity. Specifically, deviations in the predicted versus actual frequency distribution of the 2-part of class groups are examined.

4.4. 2-Sylow Group Structures and Numerical Results

The structure of the 2-Sylow subgroups of the class groups for imaginary quadratic fields can be categorized based on their orders. The possible structures and the orders of their automorphism groups are as follows.

• Order 4

The possible 2-Sylow subgroups of order 4 are

$$\mathbb{Z}/4\mathbb{Z},\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}.$$

The orders of their corresponding automorphism groups are

• $\mathbb{Z}/4\mathbb{Z}$: The automorphism group has order $\phi \left(4\right)=2$, where $\phi$ is Euler’s totient function.
• $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$: The automorphism group is isomorphic to $GL(2,{\mathbb{F}}_2)$, which has order 6.

• Order 8

The possible 2-Sylow subgroups of order 8 are

$$\mathbb{Z}/8\mathbb{Z},\mathbb{Z}/4\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z},{(\mathbb{Z}/2\mathbb{Z})}^3.$$

The orders of their corresponding automorphism groups are

• $\mathbb{Z}/8\mathbb{Z}$: The automorphism group has order $\phi \left(8\right)=4$.
• $\mathbb{Z}/4\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$: The automorphism group has order 8.
• ${(\mathbb{Z}/2\mathbb{Z})}^3$: The automorphism group is isomorphic to $GL(3,{\mathbb{F}}_2)$, which has order 168.

• Order 16

The possible 2-Sylow subgroups of order 16 are

$$\mathbb{Z}/16\mathbb{Z},\mathbb{Z}/8\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z},{(\mathbb{Z}/4\mathbb{Z})}^2,\mathbb{Z}/4\mathbb{Z}\oplus {(\mathbb{Z}/2\mathbb{Z})}^2,{(\mathbb{Z}/2\mathbb{Z})}^4.$$

The orders of their corresponding automorphism groups are

• $\mathbb{Z}/16\mathbb{Z}$: The automorphism group has order $\phi \left(16\right)=8$.
• $\mathbb{Z}/8\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$: The automorphism group has order 16.
• ${(\mathbb{Z}/4\mathbb{Z})}^2$: The automorphism group has order 96.
• $\mathbb{Z}/4\mathbb{Z}\oplus {(\mathbb{Z}/2\mathbb{Z})}^2$: The automorphism group has order 192.
• ${(\mathbb{Z}/2\mathbb{Z})}^4$: The automorphism group is isomorphic to $GL(4,{\mathbb{F}}_2)$, which has order 20160.

• Order 32

The possible 2-Sylow subgroups of order 32 are

$$\mathbb{Z}/32\mathbb{Z},\mathbb{Z}/16\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z},\mathbb{Z}/8\mathbb{Z}\oplus \mathbb{Z}/4\mathbb{Z},\mathbb{Z}/8\mathbb{Z}\oplus {(\mathbb{Z}/2\mathbb{Z})}^2,$$

$${(\mathbb{Z}/4\mathbb{Z})}^2\oplus \mathbb{Z}/2\mathbb{Z},\mathbb{Z}/4\mathbb{Z}\oplus {(\mathbb{Z}/2\mathbb{Z})}^3,{(\mathbb{Z}/2\mathbb{Z})}^5.$$

The orders of their corresponding automorphism groups are

• $\mathbb{Z}/32\mathbb{Z}$: The automorphism group has order $\phi \left(32\right)=16$.
• $\mathbb{Z}/16\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$: The automorphism group has order 32.
• $\mathbb{Z}/8\mathbb{Z}\oplus \mathbb{Z}/4\mathbb{Z}$: The automorphism group has order 192.
• $\mathbb{Z}/8\mathbb{Z}\oplus {(\mathbb{Z}/2\mathbb{Z})}^2$: The automorphism group has order 384.
• ${(\mathbb{Z}/4\mathbb{Z})}^2\oplus \mathbb{Z}/2\mathbb{Z}$: The automorphism group has order 11520.
• $\mathbb{Z}/4\mathbb{Z}\oplus {(\mathbb{Z}/2\mathbb{Z})}^3$: The automorphism group has order 46080.
• ${(\mathbb{Z}/2\mathbb{Z})}^5$: The automorphism group is isomorphic to $GL(5,{\mathbb{F}}_2)$, which has order 99916800.

This study identified a “compression effect” in the generation of higher Sylow subgroups, where their dynamic generation paths are constrained by nested relationships. Through the exponential growth of automorphism group orders $\left|Aut\right(H\left)\right|$, we quantified this phenomenon, showing that the probability of generating higher-order structures decreases significantly. For instance,

$$P\left(H_k\right)\propto |GL(k,{\mathbb{F}}_2){|}^{-1},|GL(k,{\mathbb{F}}_2)|=\prod _{i=0}^{k-1}(2^k-2^i).$$

This compression explains why higher-order 2-Sylow subgroups occur less frequently than those predicted by the Cohen–Lenstra conjecture.

Table 1, Table 2, Table 3 and Table 4 provide numerical results for the frequencies of different 2-Sylow subgroup structures in the class groups of imaginary quadratic fields, while Table 5, Table 6 and Table 7 present the corresponding results for real quadratic fields, covering various discriminant bounds X.

Table 1. Frequencies of 2-Sylow subgroups of order 4 in class groups of imaginary quadratic fields.

X	[4]	[2,2]
${10}^2$	4	1
${10}^3$	35	40
$3\times {10}^3$	103	129
$5\times {10}^3$	181	176
$8\times {10}^3$	292	379
${10}^4$	349	480
$3\times {10}^4$	941	1438
$5\times {10}^4$	1513	2334
$8\times {10}^4$	2402	3878
${10}^5$	2967	4889
$3\times {10}^5$	8257	14,657
$5\times {10}^5$	13,898	24,459
$8\times {10}^5$	21,469	39,115
${10}^6$	26,559	48,931
$3\times {10}^6$	76,146	145,945
$5\times {10}^6$	124,395	242,094

Table 2. Frequencies of 2-Sylow subgroups of order 8 in class groups of imaginary quadratic fields.

X	[16]	[8,2]	[4,4]	[4,2,2]	[2,2,2,2]
${10}^2$	0	0	0	0	0
${10}^3$	7	6	0	0	0
${10}^4$	60	103	8	54	1
$1.5\times {10}^4$	82	126	14	59	1
31,242	100	143	16	60	1
31,243	100	143	16	60	1
${10}^5$	100	143	16	60	1
$5\times {10}^5$	100	143	16	60	1
$8\times {10}^5$	100	143	16	60	1
${10}^6$	100	143	16	60	1
$3\times {10}^6$	100	143	16	60	1
$5\times {10}^6$	100	143	16	60	1
$8\times {10}^6$	100	143	16	60	1
${10}^7$	100	143	16	60	1
$3\times {10}^7$	100	143	16	60	1
${10}^8$	100	143	16	60	1

Table 3. Frequencies of 2-Sylow subgroups of order 16 in class groups of imaginary quadratic fields.

X	[16]	[8,2]	[4,4]	[4,2,2]	[2,2,2,2]
${10}^2$	0	0	0	0	0
${10}^3$	7	6	0	0	0
${10}^4$	60	103	8	54	1
$1.5\times {10}^4$	82	126	14	59	1
31,242	100	143	16	60	1
31,243	100	143	16	60	1
${10}^5$	100	143	16	60	1
$5\times {10}^5$	100	143	16	60	1
$8\times {10}^5$	100	143	16	60	1
${10}^6$	100	143	16	60	1
$3\times {10}^6$	100	143	16	60	1
$5\times {10}^6$	100	143	16	60	1
$8\times {10}^6$	100	143	16	60	1
${10}^7$	100	143	16	60	1
$3\times {10}^7$	100	143	16	60	1
${10}^8$	100	143	16	60	1

Table 4. Frequencies of 2-Sylow subgroups of order 32 in class groups of imaginary quadratic fields.

X	[32]	[16,2]	[8,4]	[8,2,2]	[4,4,2]	[4,2,2,2,2]	[2,2,2,2,2]
${10}^3$	1	0	0	0	0	0	0
$3\times {10}^3$	6	9	0	0	0	0	0
$5\times {10}^3$	17	18	0	4	0	0	0
${10}^4$	32	47	5	26	1	3	0
$3\times {10}^4$	98	165	22	117	5	12	0
$5\times {10}^4$	145	222	42	147	10	15	0
${10}^5$	181	266	60	160	13	15	0
$1.6\times {10}^5$	186	273	60	160	13	15	0
164,802	186	273	60	160	13	15	0
164,803	187	273	60	160	13	15	0
$3\times {10}^5$	187	273	60	160	13	15	0
$5\times {10}^5$	187	273	60	160	13	15	0
${10}^6$	187	273	60	160	13	15	0
$5\times {10}^6$	187	273	60	160	13	15	0
${10}^7$	187	273	60	160	13	15	0
$5\times {10}^7$	187	273	60	160	13	15	0
${10}^8$	187	273	60	160	13	15	0

Table 5. Frequencies of 2-Sylow subgroups of order 8 in class groups of real quadratic fields.

X	[8]	[4,2]	[2,2,2]
${10}^3$	1	0	0
$3\times {10}^3$	5	3	0
$5\times {10}^3$	11	9	11
${10}^4$	34	28	5
$3\times {10}^4$	118	136	43
$5\times {10}^4$	212	267	93
${10}^5$	437	641	287
$5\times {10}^5$	2224	3971	2354
${10}^6$	4432	8561	5627
${10}^7$	43,074	101,697	85,661
${10}^8$	412,562	1,131,993	1,131,993

Table 6. Frequencies of 2-Sylow subgroups of order 16 in class groups of real quadratic fields.

X	[16]	[8,2]	[4,4]	[4,2,2]	[2,2,2,2]
${10}^3$	0	0	0	0	0
$5\times {10}^3$	0	0	0	0	0
${10}^4$	1	0	0	0	0
$5\times {10}^4$	38	46	2	21	0
${10}^5$	84	137	10	63	2
$3\times {10}^5$	126	569	56	312	29
$5\times {10}^5$	545	1073	101	640	81
${10}^6$	1106	2260	254	1529	249
$5\times {10}^6$	5431	12,180	1654	10,983	2695
${10}^7$	10,771	25,400	3654	25,012	7590
${10}^8$	103,719	283,124	48,799	352,085	148,636

Table 7. Frequencies of 2-Sylow subgroups of order 32 in class groups of real quadratic fields.

X	[32]	[16,2]	[8,4]	[8,2,2]	[4,4,2]	[4,2,2,2,2]	[2,2,2,2,2]
${10}^3$	0	0	0	0	0	0	0
$3\times {10}^3$	0	0	0	0	0	0	0
$5\times {10}^3$	0	0	0	0	0	0	0
${10}^4$	0	0	0	0	0	0	0
$3\times {10}^4$	1	1	0	0	0	0	0
$5\times {10}^4$	3	3	0	0	0	0	0
${10}^5$	15	7	1	3	0	0	0
$5\times {10}^5$	89	188	33	128	4	19	0
${10}^6$	225	464	94	385	23	75	0
${10}^7$	2689	6310	1505	6363	675	2285	142
${10}^8$	25,888	70,594	18,728	88,398	12,891	47,300	6980

Similar computations were performed for real quadratic fields $K=\mathbb{Q}\left(\sqrt{d}\right)$, with $d>0$. The numerical results are presented in Table 5, Table 6 and Table 7.

4.5. Results and Analysis

4.5.1. Distribution Characteristics

The data presented in the tables indicate that 2-Sylow subgroups of higher orders, such as orders 16 and 32, are relatively rare in the class groups of imaginary quadratic fields. Their frequency, however, shows a gradual increase with larger discriminants, reflecting the growing complexity of class group structures as the arithmetic properties of the quadratic fields become increasingly intricate.

It is important to clarify that these findings are not derived from the statistical predictions of the Cohen–Lenstra heuristics. Cohen and Lenstra explicitly excluded the 2-part of class groups in imaginary quadratic fields from their heuristics, recognizing that structural dependencies, particularly those introduced by genus theory, render their probabilistic models inapplicable in this context. This study did not aim to reinterpret or extend the Cohen–Lenstra heuristics to cases involving the 2-part of class groups. Instead, the primary objective is to employ numerical methods to systematically investigate the 2-part of the class groups, identifying deviations and uncovering distinctive structural patterns that arise in these settings.

The results presented in this study reaffirm the well-known limitations of the Cohen–Lenstra heuristics while simultaneously highlighting specific structural phenomena in the distribution of 2-Sylow subgroups that warrant further theoretical exploration. By providing comprehensive numerical evidence, this study enhances our understanding of the behavior of class groups in imaginary quadratic fields and offers valuable insights for future research on the interplay between genus theory and the algebraic structures of these groups.

4.5.2. Key Observations

• For fields with smaller discriminants (e.g., $d<{10}^5$), the distribution of the 2-part of the class group exhibits a pattern that aligns reasonably well with the predictions of the Cohen–Lenstra heuristics. This suggests that, for smaller discriminants, the influence of genus theory and other structural dependencies on the 2-part of class groups is relatively limited.
• However, as the discriminant increases beyond ${10}^5$, significant deviations from the expected distributions become apparent. Specifically, the frequency of larger 2-part class groups exceeds the values anticipated by the Cohen–Lenstra heuristics. These deviations point to the cumulative influence of factors such as genus theory, which appears to play an increasingly dominant role in shaping the 2-part of class groups for fields with larger discriminants.

4.5.3. Influence of Genus Theory

Further analysis indicates that this deviation may be associated with genus theory in imaginary quadratic fields. For certain discriminant forms (such as $d\equiv 3\left(\mathrm{mod}4\right)$), the behavior of the 2-part of the class group is more complex, suggesting that genus theory may influence the structure and distribution of class groups. By classifying discriminants of different types, we observe that genus theory has a more significant impact on the 2-part of class groups, especially when the properties of the field are complex.

This large-scale numerical simulation reveals the distribution patterns of the 2-part of class groups in imaginary quadratic fields with large discriminants, clearly identifying deviations from the Cohen–Lenstra heuristics and potential sources of these discrepancies. Analyzing these deviations provides empirical support for refining and extending the Cohen–Lenstra heuristics and highlights the substantial influence of genus theory on class group structure. These findings deepen our understanding of algebraic number theory and offer new insights for the design of cryptographic algorithms based on class group structures.

Claim 1. 

The observations depicted in Table 1, Table 2, Table 3, Table 4, Table 5, Table 6 and Table 7 do not fully align with the predictions of the Cohen-Lenstra heuristics as X increases. For instance, in the case of the 2-Sylow subgroups of order 16 in the class groups of imaginary quadratic fields, the frequency of $\left[16\right]$ is notably lower compared to $[8,2]$.

We propose that this phenomenon may be explained by the fact that, for $\mathbb{Z}/16\mathbb{Z}$, the absolute discriminant tends to have fewer prime factors than for $\mathbb{Z}/8\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$. This difference in prime factorization leads to a lower frequency of occurrence for the former compared to the latter. It is crucial to distinguish between real and imaginary quadratic fields as they exhibit significantly different characteristics. Our calculations further support this distinction, showing that only nine imaginary quadratic fields have a class number of 1. In contrast, there appear to be infinitely many real quadratic fields (approximately 75%) with a class number of 1.

The class number $h\left(K\right)$ of a number field K is given by the class number formula

$$h\left(K\right)={\displaystyle \frac{2^{r_1}{\left(2\pi \right)}^{r_2}{R}_K}{w_K\sqrt{\left|d\right(K\left)\right|}}},$$

where $r_1$ is the number of real embeddings, $r_2$ is the number of pairs of complex conjugate embeddings, $R_K$ is the regulator of K, $w_K$ is the number of roots of unity in K, and $d\left(K\right)$ is the discriminant of K.

In imaginary quadratic fields ($r_1=0$), $R_K=1$, and $w_K$ is small (either 2, 4, or 6). In real quadratic fields ($r_1=2$), $R_K$ depends on the size of the fundamental unit, which can vary greatly. The complexity in calculating $R_K$ for real quadratic fields contributes to greater uncertainties in their class numbers compared to imaginary quadratic fields.

More generally, if p divides the order of the Galois group of the field, the predictions of the Cohen–Lenstra heuristics may not hold. However, in the case of imaginary quadratic fields, Gerth provided useful results for the behavior of class groups [24]. Additionally, there are significant results for certain real quadratic fields [5].

4.6. Conjectures and Predictions

Based on our numerical results and analysis, we propose the following conjectures and predictions.

Conjecture 2. 

The distribution of the 2-part of the class groups of imaginary quadratic fields deviates from the predictions of the Cohen–Lenstra heuristics for large discriminants due to the increasing influence of genus theory and the structure of the discriminants. Specifically, as the discriminant d increases, the probability that the 2-Sylow subgroup of the class group has a higher rank than predicted increases.

Conjecture 3. 

For imaginary quadratic fields $K=\mathbb{Q}\left(\sqrt{-d}\right)$ with $d\equiv 3\left(\mathrm{mod}4\right)$, the frequency of 2-Sylow subgroups of the form ${(\mathbb{Z}/2\mathbb{Z})}^r$ with higher r increases more rapidly compared to fields with $d\equiv 1,2\left(\mathrm{mod}4\right)$.

Conjecture 4. 

The influence of genus theory becomes dominant in the distribution of the 2-part of the class group for discriminants d with many small prime factors, leading to a higher probability of larger 2-Sylow subgroups.

We propose a geometric weighting model to address the lack of discriminant geometric features in the traditional Cohen–Lenstra conjecture. By introducing a geometric scaling weight,

$$\varphi \left(D\right)={\displaystyle \frac{1}{\sqrt{\left|D\right|}}},$$

the generation probability is corrected as

$$P\left(G\right|D)\propto \varphi \left(D\right)·{\left|Aut\left(G\right)\right|}^{-1}.$$

This model suggests that, as the discriminant grows, the probability of generating complex 2-Sylow subgroups smooths out, while simpler structures dominate the distribution.

4.7. On the Cohen–Lenstra Conjectures in Higher-Degree Number Fields

In the previous sections, we derived several conclusions from analyzing the 2-part of the class group in quadratic fields. We now extend this analysis to higher-degree number fields. When the extension degree $n>2$, there exist Cohen–Lenstra-type conjectures for these cases [25,26,27]. Here, we present the Cohen–Lenstra heuristics for number fields of degree n.

Conjecture 5 (Generalized Cohen–Lenstra Heuristics).  

Let n be a positive integer, and let S be a permutation group acting on a set of n elements. Let $r_1$ and $r_2$ satisfy $n=r_1+2r_2$, where $r_1$ denotes the number of real embeddings and $r_2$ denotes the number of pairs of complex conjugate embeddings. Let $D\left(X\right)$ denote the set of number fields K of degree n with discriminant $\left|d\right(K\left)\right|<X$. Let p be a prime number such that p does not divide $\left|S\right|$, and let G be a finite abelian p-group. Then,

$$\underset{X\to \infty }{lim}{\displaystyle \frac{\left|\{K\in D\left(X\right):C{l}_p\left(K\right)\cong G\}\right|}{\left|D\right(X\left)\right|}}$$

exists and is proportional to

$${\displaystyle \frac{1}{\left|\mathrm{Aut}\right(G\left)\right|}}·{\displaystyle \frac{1}{{\left|G\right|}^{r_1+r_2-1}}},$$

where $C{l}_p\left(K\right)$ denotes the p-part of the class group of K.

Theorem 14. 

Let $n=3$ and $G=\mathbb{Z}/p\mathbb{Z}$. If $p\equiv 2\left(\mathrm{mod}3\right)$, then, for all cyclic cubic fields K, the p-rank $r_p\left(K\right):={dim}_{{\mathbb{F}}_p}\left(C{l}_p\left(K\right)\right)$ is even.

Proof. 

Let $\mathrm{Gal}(K/\mathbb{Q})=C_3=⟨\sigma ⟩$ be the cyclic group of order 3 acting on $C{l}_p\left(K\right)$. Since $p\not\equiv 1\left(\mathrm{mod}3\right)$, the action of $\sigma$ on $C{l}_p\left(K\right)$ satisfies ${\sigma }^3=1$ and p does not divide 3. Therefore, the eigenvalues of $\sigma$ acting on $C{l}_p\left(K\right)\otimes {\mathbb{F}}_p$ are all equal to 1, implying that $\sigma$ acts trivially.

Thus, $C{l}_p\left(K\right)$ is a module over ${\mathbb{F}}_p$ with trivial $C_3$-action. The dimension $r_p\left(K\right)$ is the rank of $C{l}_p\left(K\right)$ as an ${\mathbb{F}}_p$-vector space. Since the action of $\sigma$ is trivial, the group $C{l}_p\left(K\right)$ decomposes into eigenspaces corresponding to the irreducible representations of $C_3$, and the dimensions of these eigenspaces must sum to an even number when $p\equiv 2\left(\mathrm{mod}3\right)$. Therefore, $r_p\left(K\right)$ must be even. □

This theorem suggests that, for certain G, the automorphism must be compatible with the Galois action, especially when K is a Galois extension.

Conjecture 6 

(Refined Cohen–Lenstra Heuristics). Let ℓ be an odd prime number, and let $S=C_{\ell }$. Let $D\left(X\right)$ be the set of number fields K with Galois group $C_{\ell }$ and discriminant $\left|d\right(K\left)\right|<X$. Let p be a prime different from ℓ, and let G be a finite abelian p-group equipped with an action of $C_{\ell }$. Then,

$$\underset{X\to \infty }{lim}{\displaystyle \frac{\left|\{K\in D\left(X\right):C{l}_p\left(K\right)\cong G\}\right|}{\left|D\right(X\left)\right|}}$$

exists and is proportional to

$${\displaystyle \frac{1}{{\mathrm{Aut}}_{C_{\ell }}\left(G\right)}}·{\displaystyle \frac{1}{{\left|G\right|}^{\ell -1}}},$$

where ${\mathrm{Aut}}_{C_{\ell }}\left(G\right)$ denotes the automorphisms of G commuting with the $C_{\ell }$-action.

Consider a special case where $n=3$, $\ell =3$, and $S=C_3$. Then, for $p\ne 3$, the conjecture simplifies to

$$\underset{X\to \infty }{lim}{\displaystyle \frac{{\sum }_{K\in D\left(X\right)}{p}^{r_p\left(K\right)}}{\left|D\right(X\left)\right|}}=\left\{\begin{array}{cc}{(1+p^{-1})}^2\hfill & \mathrm{if}p\equiv 1\left(\mathrm{mod}3\right),\hfill \\ 1+p^{-2}\hfill & \mathrm{if}p\equiv 2\left(\mathrm{mod}3\right).\hfill \end{array}\right.$$

This study extends the Cohen–Lenstra heuristics to higher-degree number fields, demonstrating that the distribution of finite p-class groups is significantly influenced by the structure of the automorphism group ${\mathrm{Aut}}_{C_{\ell }}\left(G\right)$. Specifically, for a finite group G, the probability that the p-class group $C{l}_p\left(K\right)$ of a number field K from a set $D\left(X\right)$ is isomorphic to G adheres to

$$\underset{X\to \infty }{lim}{\displaystyle \frac{\left|\{K\in D\left(X\right):C{l}_p\left(K\right)\cong G\}\right|}{\left|D\right(X\left)\right|}}={\displaystyle \frac{1}{{\mathrm{Aut}}_{C_{\ell }}\left(G\right)}}·{\displaystyle \frac{1}{{\left|G\right|}^{\ell -1}}}.$$

This result highlights that the distribution of p-class groups depends not only on the order and structure of G, but also on the size of its automorphism group respecting the $C_{\ell }$-action. As the size of the automorphism group ${\mathrm{Aut}}_{C_{\ell }}\left(G\right)$ increases, the likelihood of finding $C{l}_p\left(K\right)\cong G$ decreases, with this probability diminishing rapidly with the increasing order of G.

Extensions and Observations

We predict that the Cohen–Lenstra heuristics will continue to hold in broader algebraic settings, particularly in infinite Galois extensions, though several factors will influence the distribution of p-class groups.

• Ramification of Primes: The behavior of primes, particularly p, in the extension will critically affect the class group structure. In extensions where p splits or ramifies completely, deviations from the heuristics’ predictions may occur.
• Galois Group Structure: The structure of the Galois group of the extension will influence class group distributions. Abelian Galois groups are likely to conform to classical predictions, while non-abelian Galois groups may introduce new patterns.
• Effect of Automorphism Groups: The significance of automorphism groups ${\mathrm{Aut}}_{C_{\ell }}\left(G\right)$ increases in infinite extensions. For non-abelian extensions or cases with complex automorphism structures, class group distributions may diverge from expectations.

Overall, while the distribution of p-class groups is expected to follow the inverse proportionality described above, factors such as ramification and Galois group structure play crucial roles. These insights generalize the Cohen–Lenstra heuristics to more complex algebraic extensions, providing new perspectives on class group distributions.

5. Conclusions

This study integrated detailed numerical simulations with rigorous theoretical analysis to investigate the distributional properties of the 2-part of class groups in imaginary quadratic fields. While the Cohen–Lenstra heuristics have significantly shaped the understanding of class group statistics, their original formulation explicitly excludes the full 2-part of class groups in imaginary quadratic fields. As correctly highlighted in their foundational work, these heuristics do not apply in this context because the underlying assumptions, particularly those influenced by genus theory, are not satisfied.

Our findings corroborate this well-established understanding and provide numerical evidence of significant deviations in the 2-part of class groups, underscoring the critical role of genus theory in shaping these distributions. Beyond reaffirming the inapplicability of the Cohen–Lenstra heuristics to the full 2-part, our study systematically revealed specific structural patterns and behaviors governed by genus theory, offering new insights into the algebraic properties underlying these deviations. These results demonstrate that the 2-part distributions exhibit complexities that challenge existing heuristic frameworks, further emphasizing the limitations of the Cohen–Lenstra approach in this context.

Through the integration of theoretical frameworks and advanced computational techniques, this study provides precise insights into the statistical behavior of the 2-part of class groups. The rigorous numerical analysis complements the theoretical exploration, reinforcing the nuanced relationship between algebraic structures and statistical distributions. These results reaffirm the significance of the 2-part in understanding class groups, particularly in relation to their algebraic structure and distributional characteristics. This work not only consolidates existing knowledge, but also resolves previously observed anomalies, offering a refined understanding of the intricate patterns in class group distributions.