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On the Galois structure of units in totally real $p$-rational number fields

Zakariae Bouazzaoui, Donghyeok Lim

TL;DR

The paper develops a comprehensive framework linking the Galois module structure of unit lattices to arithmetic invariants via factor equivalence and regulator constants. It proves a general criterion: for a Galois extension $L/k$ unramified at infinity, $E_L$ is factor equivalent to $\\mathcal{A}_G$ iff a product of powers of class-number-related terms equals one for all $G$-relations, extending Burns’ results to non-abelian groups. It then shows non-existence of Minkowski units in non-abelian $p$-rational $p$-extensions of $\\mathbb{Q}$ and establishes finite-step rigidity results for relative Galois module structures over totally real $p$-rational fields, with explicit decomposition patterns into a fixed finite set of lattices. Collectively, these results reveal deep connections between regulator constants, class numbers, and Galois-module decompositions, while providing practical criteria and finite classifications for wide families of totally real fields and their extensions. The findings have implications for understanding Minkowski-type phenomena, pro-$p$ towers, and the arithmetic of totally real $p$-rational fields in the context of Galois module theory.

Abstract

The theory of factor-equivalence of integral lattices establishes a far-reaching relationship between the Galois module structure of the unit group of the ring of integers of a number field and its arithmetic. For a number field $K$ that is Galois over $\mathbb{Q}$ or an imaginary quadratic field, we prove a necessary and sufficient condition on the quotients of class numbers of subfields of $K$, for the quotient $E_{K}$ of the unit group of the ring of integers of $K$ modulo the subgroup of roots of unity to be factor equivalent to the standard cyclic Galois module. Using strong arithmetic properties of totally real $p$-rational number fields, we prove that the non-abelian $p$-rational $p$-extensions of $\mathbb{Q}$ do not admit Minkowski units, thereby extending a result of Burns to non-abelian number fields. We also study the relative Galois module structure of $E_{L}$ for varying Galois extensions $L/F$ of totally real $p$-rational number fields whose Galois groups are isomorphic to a fixed finite group $G$. In that case, we prove that there exists a finite set $Ω$ of $\mathbb{Z}_p[G]$-lattices such that for every $L$, $\mathbb{Z}_{p} \otimes_{\mathbb{Z}} E_{L}$ is factor equivalent to $\mathbb{Z}_{p}[G]^{n} \oplus X$ as $\mathbb{Z}_p[G]$-lattices for some $X \in Ω$ and an integer $n \geq 0$.

On the Galois structure of units in totally real $p$-rational number fields

TL;DR

The paper develops a comprehensive framework linking the Galois module structure of unit lattices to arithmetic invariants via factor equivalence and regulator constants. It proves a general criterion: for a Galois extension unramified at infinity, is factor equivalent to iff a product of powers of class-number-related terms equals one for all -relations, extending Burns’ results to non-abelian groups. It then shows non-existence of Minkowski units in non-abelian -rational -extensions of and establishes finite-step rigidity results for relative Galois module structures over totally real -rational fields, with explicit decomposition patterns into a fixed finite set of lattices. Collectively, these results reveal deep connections between regulator constants, class numbers, and Galois-module decompositions, while providing practical criteria and finite classifications for wide families of totally real fields and their extensions. The findings have implications for understanding Minkowski-type phenomena, pro- towers, and the arithmetic of totally real -rational fields in the context of Galois module theory.

Abstract

The theory of factor-equivalence of integral lattices establishes a far-reaching relationship between the Galois module structure of the unit group of the ring of integers of a number field and its arithmetic. For a number field that is Galois over or an imaginary quadratic field, we prove a necessary and sufficient condition on the quotients of class numbers of subfields of , for the quotient of the unit group of the ring of integers of modulo the subgroup of roots of unity to be factor equivalent to the standard cyclic Galois module. Using strong arithmetic properties of totally real -rational number fields, we prove that the non-abelian -rational -extensions of do not admit Minkowski units, thereby extending a result of Burns to non-abelian number fields. We also study the relative Galois module structure of for varying Galois extensions of totally real -rational number fields whose Galois groups are isomorphic to a fixed finite group . In that case, we prove that there exists a finite set of -lattices such that for every , is factor equivalent to as -lattices for some and an integer .
Paper Structure (17 sections, 49 theorems, 85 equations)

This paper contains 17 sections, 49 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Genus equivalence
  3. Factor equivalence, theorems of Burns and regulator constant
  4. Factor equivalence of integral lattices
  5. Theorems of Burns
  6. Arithmetic Criteria for the Existence of Local Minkowski Units in Abelian Extensions
  7. Applications of the Arithmetic Criteria to Genus Field Extensions
  8. Factor equivalence and regulator constants
  9. Basic facts on regulator constants
  10. Theorems of Bartel
  11. Regulator constants of some standard lattices
  12. Proof of Theorem \ref{['maintheorem']}
  13. The arithmetic properties of totally real $p$-rational number fields
  14. Totally real $p$-rational number fields
  15. On the $p$-class numbers in a pro-$p$ tower of totally real $p$-rational number fields when $p$ does not split
  16. ...and 2 more sections

Key Result

Theorem A

Let $K$ be a number field that is Galois over an admissible field $k$ with Galois group $G_{K/k}$. Assume that $K/k$ is unramified at the infinite places of $k$. For each subgroup $H$ of $G_{K/k}$, let $h_{K^{H}}$ (resp. $w_{K^{H}}$) denote the class number (resp. the number of roots of unity) of th induced by the embedding $\mu(K) \hookrightarrow \mathcal{O}_{K}^{\times}$. Then, $E_K$ is factor e

Theorems & Definitions (104)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 2.1
  • Theorem 2.2
  • Example 2.3
  • Remark 2.4
  • Definition 3.1
  • Proposition 3.2
  • ...and 94 more