On a characterisation of perfectoid fields by Iwasawa theory

TL;DR

The paper proves that for a nontrivial de Rham $p$-adic Galois representation $V$ of $G_K$ whose Hodge–Tate weights are not all nonpositive, the vanishing of the universal-norm Iwasawa module $\mathrm H^1_{\mathrm{Iw},g}(K,L,T)$ characterises when the completion $\hat L$ is perfectoid. The argument weaves together topological continuous group cohomology, $p$-adic period rings, and Bloch–Kato groups, using the fundamental exact sequence and comparison maps to relate the vanishing of universal norms to perfectoid-ness. It generalizes results of Coates–Greenberg and Bondarko by treating de Rham representations beyond abelian varieties and $p$-divisible groups, and provides a framework in which the (non-)perfectoid nature of $\hat L$ is detected by the Galois-cohomological behavior of $V$ via $\mathrm H^1_{\mathrm{Iw},g}(K,L,T)$. The work also yields concrete corollaries for abelian varieties and $p$-divisible groups and emphasizes the necessity of the HT-weight hypothesis through explicit examples. Overall, the results offer a precise bridge between Iwasawa-theoretic invariants and the Fontaine–Scholze paradigm of perfectoid fields, with potential impact on understanding Galois-cohomological phenomena in nonperfectoid 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.

Theorems & Definitions (136)

• Theorem 1.2.1
• Remark 1.2.2
• Remark 1.2.3
• Remark 1.2.4
• Proposition 2.1.1
• proof
• Remark 2.2.1
• Remark 2.2.3
• Proposition 2.3.1
• proof