Laurent Fargues (IAS and Paris-Sud)

Reduction Theory for p-Adic Moduli Spaces of Abelian Varieties and p-Divisible Groups

I will first explain how one can parametrize some particular cases of p-adic moduli spaces of p-divisible groups, Lubin-Tate spaces, by the geometric realization of Bruhat-Tits buildings. By a p-adic space I mean Berkovich analytic space. I will then explain how this extends to a parameterization of the p-adic spaces associated to some particular cases of Shimura varieties by compactifications of those buildings, the Lubin-Tate spaces being some p-adic Milnor fiber inside those p-adic Shimura varieties. I will then explain how to study the p-adic geometry of more general Shimura varieties or more general moduli spaces of p-divisible groups by using Harder-Narasimhan filtrations for finite flat group schemes and p-divisible groups. Those filtrations allows us to study p-adic reduction theory (in the sens of Siegel) for the action of Hecke correspondences at p on those p-adic moduli spaces.