On Drinfeld's representability theorem

TL;DR

This paper provides a clearer proof of Drinfeld's representability theorem and offers a detailed presentation of the associated moduli space and formal model of the $p$-adic symmetric space.

Contribution

It introduces a new, more transparent proof of Drinfeld's theorem and elaborates on the structure of the moduli space and formal models involved.

Findings

New proof simplifies understanding of Drinfeld's theorem

Detailed description of Drinfeld's moduli space

Explicit construction of the formal model of the $p$-adic symmetric space

Abstract

In the seventies, V. G. Drinfeld proved that a moduli problem of deformations by quasi-isogenies of certain $p$-divisible groups with extra actions is representable by an explicit semi-stable model of the $p$-adic symmetric space. This theorem, known as \emph{Drinfeld's representability theorem}, has been one of the cornerstones of geometric aspects in $p$-adic Hodge theory. The purpose of these notes is twofold. On the one hand we give a new and more transparent proof of Drinfeld's representability theorem; on the other hand, we give a detailed presentation of Drinfeld's moduli space and the formal model of the $p$-adic symmetric space.