Group schemes of period p>2

TL;DR

This paper provides a direct proof of Breuil's classification of finite flat group schemes killed by p over p-adic valuation rings for primes p>2, and relates their Galois modules via the field-of-norms functor.

Contribution

It offers a new direct proof of Breuil's classification and links Galois modules of these group schemes with their characteristic p analogues using Fontaine-Wintenberger theory.

Findings

Confirmed the classification of finite flat group schemes for p>2

Established an isomorphism of Galois modules via the field-of-norms functor

Extended understanding of group schemes in p-adic and characteristic p contexts

Abstract

For a prime number p>2, we give a direct proof of Breuil's classification of killed by p finite flat group schemes over the valuation ring of a p-adic field with perfect residue field. As application we prove that the Galois modules of geometric points of such group schemes and of their characteristic p analogues coming from Faltings's strict modules can be identified via the Fontaine-Wintenberger field-of-norms functor.