Adams operations and Galois structure

TL;DR

This paper introduces a new method leveraging Adams operations and equivariant K-theory to analyze Galois module structures of cohomology on varieties, proving two conjectures for curves.

Contribution

It develops a novel Adams-Riemann-Roch type theorem combining Kunneth formula, localization, and cyclotomic field results, advancing Galois module theory.

Findings

Proves two conjectures of Chinburg-Pappas-Taylor for curves.

Provides a new computational approach for Galois module structures.

Introduces a novel Adams-Riemann-Roch theorem in equivariant K-theory.

Abstract

We present a new method for determining the Galois module structure of the cohomology of coherent sheaves on varieties over the integers with a tame action of a finite group. This uses a novel Adams-Riemann-Roch type theorem obtained by combining the Kunneth formula with localization in equivariant K-theory and classical results about cyclotomic fields. As an application, we show two conjectures of Chinburg-Pappas-Taylor, in the case of curves.