Tate modules of universal p-divisible groups

TL;DR

This paper studies the Galois representations arising from Tate modules of universal p-divisible groups, providing explicit descriptions of their images in various deformation settings using stratification techniques.

Contribution

It determines the image of Galois representations for universal deformations of p-divisible groups in mixed and positive characteristic, extending known results to more complex cases.

Findings

Explicit description of Galois image for universal deformations

Reduction to one-dimensional cases via Newton polygon stratification

Results applicable to bi-infinitesimal and infinitesimal groups

Abstract

A p-divisible group over a complete local domain determines a Galois representation on the Tate module of its generic fibre. We determine the image of this representation for the universal deformation in mixed characteristic of a bi-infinitesimal group and for the p-rank strata of the universal deformation in positive characteristic of an infinitesimal group. The method is a reduction to the known case of one-dimensional groups by a deformation argument based on properties of the stratification by Newton polygons.