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Equidistribution of realizable Steinitz classes for cyclic Kummer extensions

Brody Lynch

TL;DR

The paper proves that the Steinitz classes of $\\mathbb{Z}/\\ell\\mathbb{Z}$-extensions $L/K$ with $K$ containing $\\zeta_{\\ell}$ are equidistributed among realizable classes in $\\mathrm{Cl}(K)$ when ordered by relative discriminant. A constructive parametrization of $\\ell$-Kummer extensions via data $(u,\\mathcal{R},\\mathfrak{Q},\\mathfrak{I})$ leads to a precise Steinitz-class formula in terms of the discriminant factors, notably $\\Delta_{L/K}$ and its $\\ell$-factor $\\mathfrak L_{L/K}$. The key technical step is showing that the densities for having a fixed $\\mathfrak L_{L/K}$ are given by an explicit, $\\mathcal{C}$-independent factor $\\rho_{\\mathfrak B}$, and that the congruence and divisibility constraints distribute uniformly across the relevant arithmetic data. Summing over finitely many $\\mathfrak L_{L/K}$ values yields the global equidistribution: for any realizable class $\\mathcal{C}$ in $R_K$, the proportion of extensions with Steinitz class $\\mathcal{C}$ tends to $1/|R_K|$, significantly strengthening and generalizing previous tame-ramification results. The approach is constructive and avoids tameness assumptions, extending known results for $\\ell=2$ and offering a framework for broader abelian Galois groups.

Abstract

Let $\ell$ be prime, and $K$ be a number field containing the $\ell$-th roots of unity. We use classical algebraic number theory and some analytic techniques to prove that the Steinitz classes of $\mathbb Z/\ell\mathbb Z$ extensions of $K$ ordered by relative discriminant are equidistributed among realizable classes in the ideal class group of $K$. For $\ell = 2$, this was proved by Kable and Wright using the deep theory of prehomogeneous vector spaces. Foster proved that Steinitz classes are uniformly distributed between realizable classes for tamely ramified elementary-$m$ extensions using the theory of Galois modules; our approach eliminates this tameness hypothesis.

Equidistribution of realizable Steinitz classes for cyclic Kummer extensions

TL;DR

The paper proves that the Steinitz classes of -extensions with containing are equidistributed among realizable classes in when ordered by relative discriminant. A constructive parametrization of -Kummer extensions via data leads to a precise Steinitz-class formula in terms of the discriminant factors, notably and its -factor . The key technical step is showing that the densities for having a fixed are given by an explicit, -independent factor , and that the congruence and divisibility constraints distribute uniformly across the relevant arithmetic data. Summing over finitely many values yields the global equidistribution: for any realizable class in , the proportion of extensions with Steinitz class tends to , significantly strengthening and generalizing previous tame-ramification results. The approach is constructive and avoids tameness assumptions, extending known results for and offering a framework for broader abelian Galois groups.

Abstract

Let be prime, and be a number field containing the -th roots of unity. We use classical algebraic number theory and some analytic techniques to prove that the Steinitz classes of extensions of ordered by relative discriminant are equidistributed among realizable classes in the ideal class group of . For , this was proved by Kable and Wright using the deep theory of prehomogeneous vector spaces. Foster proved that Steinitz classes are uniformly distributed between realizable classes for tamely ramified elementary- extensions using the theory of Galois modules; our approach eliminates this tameness hypothesis.

Paper Structure

This paper contains 12 sections, 27 theorems, 93 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Outline of proof
  3. Preliminary results on $\ell$-Kummer extensions
  4. Parametrizing $\ell$-Kummer extensions
  5. Distribution of $\ell$-factors of the discriminant
  6. Congruence Condition
  7. Divisibility Condition
  8. Existence and independence of $\rho_{\mathfrak B}$
  9. Formula for Steinitz class
  10. Equidistribution of Steinitz Classes
  11. Acknowledgments
  12. Notation Index

Key Result

Theorem 1.1

Let $K$ be a number field and $G$ be a finite abelian group. Also, let $\mathcal{C} \in \mathop{\mathrm{Cl}}\nolimits(K)$ be the square of a realizable ideal class. Then then there exists a positive number $c(K, G, \mathcal{C})$ such that as $X \rightarrow \infty$ for some constants $a$ and $b$ depending on $G$ and $K$.

Figures (1)

  • Figure 1: $\ell$-Kummer extensions $L/K$ partitioned by possible values of $\mathfrak L_{L/K}$ and by $\mathop{\mathrm{St}}\nolimits(\mathcal{O}_L)$

Theorems & Definitions (51)

  • Theorem 1.1: Bekyel Bekyel
  • Theorem 1.2: Kable-Wright KW, Bhargava-Shankar-Wang Bhargava
  • Theorem 1.3: Foster Foster
  • Theorem 1.4: Bekyel Bekyel
  • Theorem 1.5
  • Proposition 2.1
  • Remark 2.2
  • Theorem 3.1: Daberkow D
  • Lemma 3.2
  • proof
  • ...and 41 more