Galois modules, ideal class groups and cubic structures

TL;DR

This paper explores the relationship between cyclotomic ideal class groups and Galois modules, revealing connections to Bernoulli numbers, K-groups, and the Kummer-Vandiver conjecture through hypercubic structures.

Contribution

It introduces a new framework linking Galois module theory, ideal class groups, and hypercubic structures, advancing understanding of cohomology and basis existence.

Findings

An invariant obstructing normal integral bases is divisible by Bernoulli numbers and K-groups.

Existence of normal bases relates to the Kummer-Vandiver conjecture.

Hyper cubic structures are key tools in the analysis.

Abstract

We establish a connection between the theory of cyclotomic ideal class groups and the theory of "geometric" Galois modules and obtain results on the Galois module structure of coherent cohomology groups of Galois covers of varieties over Z. In particular, we show that an invariant that measures the obstruction to the existence of a virtual normal integral basis for the coherent cohomology of such covers is annihilated by a product of certain Bernoulli numbers with orders of even K-groups of Z. We also show that the existence of such a normal integral basis is closely connected to the truth of the Kummer-Vandiver conjecture for the prime divisors of the degree of the cover. Our main tool is a theory of "hypercubic structures" for line bundles over group schemes.