Index formulae for integral Galois modules

TL;DR

This paper develops general index formulae for integral Galois modules, connecting their structure to arithmetic invariants across various algebraic objects over number fields, extending and unifying previous results.

Contribution

It generalizes known index formulae for units to higher K-groups and Mordell-Weil groups, covering more Galois groups and unifying different approaches.

Findings

Derived index formulae linking Galois module structure to class numbers and Tamagawa numbers.

Extended known results from units to higher K-groups and elliptic curves.

Unified various approaches in the theory of Galois modules.

Abstract

We prove very general index formulae for integral Galois modules, specifically for units in rings of integers of number fields, for higher K-groups of rings of integers, and for Mordell-Weil groups of elliptic curves over number fields. These formulae link the respective Galois module structure to other arithmetic invariants, such as class numbers, or Tamagawa numbers and Tate-Shafarevich groups. This is a generalisation of known results on units to other Galois modules and to many more Galois groups, and at the same time a unification of the approaches hitherto developed in the case of units.