On a characterisation of perfectoid fields by Iwasawa theory

TL;DR

This paper characterizes perfectoid fields via the vanishing of universal norms related to certain Galois representations, extending previous results on abelian varieties and p-divisible groups.

Contribution

It generalizes existing characterizations of perfectoid fields using Iwasawa theory and Galois representations with specific Hodge-Tate weights.

Findings

Vanishing of universal norms characterizes perfectoid fields.

Generalization of Coates and Greenberg's results.

Extension of Bondarko's results to broader contexts.

Abstract

We prove that the vanishing of the module of universal norms associated with a de Rham Galois representation whose Hodge-Tate weights are not all non-positive characterises the algebraic extensions of the field of $p$-adic numbers whose completion is a perfectoid field. We thereby generalise results by Coates and Greenberg for abelian varieties, and by Bondarko for $p$-divisible groups.