Regulator Constants and Cohomology

TL;DR

This paper establishes a connection between regulator constants of modules over group rings and group cohomology, with implications for number theory and algebraic structures like units, class groups, and elliptic curves.

Contribution

It introduces a novel relationship between regulator constants and G-cohomology, applying this to modules in number theory such as rings of integers and K-theory groups.

Findings

Relation between regulator constants and G-cohomology established

Implications for number theoretic modules like units and Mordell-Weil groups

Provides new tools for studying algebraic structures in number theory

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.