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Regulator Constants and Cohomology

Luca Caputo

Abstract

We show how regulator constants of a finitely generated $\mathbb{Z}[G]$-module can be related to $G$-cohomology, where $G$ is a finite group. We then derive consequences of such relation for modules naturally arising in number theory, such as ring of integers and units of number fields, $K$-theory groups of ring of integers and Mordell-Weil groups of elliptic curves.

Regulator Constants and Cohomology

Abstract

We show how regulator constants of a finitely generated -module can be related to -cohomology, where is a finite group. We then derive consequences of such relation for modules naturally arising in number theory, such as ring of integers and units of number fields, -theory groups of ring of integers and Mordell-Weil groups of elliptic curves.
Paper Structure (7 sections, 18 theorems, 133 equations)

This paper contains 7 sections, 18 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Regulator constants of dual modules
  3. The dihedral case
  4. Arithmetic applications
  5. Ring of integers of number fields
  6. Units of number fields
  7. Elliptic curves

Key Result

Proposition 2.2

Let $\phi:P_1\to P_2$ be any injective homomorphism with finite cokernel. Then, for any finitely generated $\mathbb{Z}[G]$-module $M$, we have

Theorems & Definitions (40)

  • Definition 2.1
  • Proposition 2.2: Bartel-de Smit
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • ...and 30 more