The classification of $p$-divisible groups over 2-adic discrete valuation rings

TL;DR

This paper classifies p-divisible groups over 2-adic valuation rings using Frobenius modules, establishing compatibility with crystalline Dieudonné theory and Galois representations, thus advancing understanding in p-adic arithmetic geometry.

Contribution

It introduces a new classification method for p-divisible groups over 2-adic rings via Frobenius modules, differing from Lau's display theory, and confirms compatibility with Galois representations.

Findings

Classification of p-divisible groups over 2-adic rings achieved

Compatibility with crystalline Dieudonné theory established

Compatibility with Galois representations demonstrated

Abstract

Let $\mathscr{O}_K$ be a 2-adic discrete valuation ring with perfect residue field $k$. We classify $p$-divisible groups and $p$-power order finite flat group schemes over $\mathscr{O}_K$ in terms of certain Frobenius module over $\mathfrak{S}:=W(k)[[u]]$. We also show the compatibility with crystalline Dieudonn\'e theory and associated Galois representations. Our approach differs from Lau's generalization of display theory, and we additionally obtain the the compatibility with associated Galois representations.