On Drinfeld's representability theorem

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.