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Construction of a $p$-extension of number fields whose unit group has prescribed Galois module structure

Takenori Kataoka, Manabu Ozaki

Abstract

Let $G$ be a finite $p$-group. We construct a $G$-extension $K/k$ of number fields such that the $p$-adic completion of the unit group of $K$ has a prescribed $\mathbb{Z}_p[G]$-module structure, up to free direct summands.

Construction of a $p$-extension of number fields whose unit group has prescribed Galois module structure

Abstract

Let be a finite -group. We construct a -extension of number fields such that the -adic completion of the unit group of has a prescribed -module structure, up to free direct summands.
Paper Structure (8 sections, 9 theorems, 47 equations)

This paper contains 8 sections, 9 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. The work of Ritter--Weiss
  3. Translation functor à la Gruenberg
  4. Tate sequence of global units
  5. Proof of the main theorem
  6. Variants
  7. First variant
  8. Second variant

Key Result

Theorem 1.1

Let $p$ be an odd prime number and $G$ a finite $p$-group. Let $C$ be a $\mathbb Z_p[G]$-lattice, namely, a finitely generated $\mathbb Z_p[G]$-module that is $\mathbb Z_p$-free. We assume that $(C\otimes_{\mathbb Z_p} \mathbb Q_p) \oplus \mathbb Q_p$ is a free $\mathbb Q_p[G]$-module. Then there ex as $\mathbb Z_p[G]$-modules for a certain $s\ge 0$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • Remark 2.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 11 more