Monodromy and irreducibility of Igusa varieties

Abstract: Igusa varieties are smooth varieties in characteristic p arising naturally as étale covers of certain subvarieties (central leaves) of Shimura varieties, for example of the ordinary locus of the modular curve. Igusa varieties over the (mu)-ordinary locus of a Shimura variety are used to define p-adic families of  automorphic forms, for instance in work of Hida;  a crucial geometric input to the study of such p-adic families is the irreducibility of Igusa varieties. In this talk I will discuss recent joint work with Luciena Xiao Xiao where we determine the irreducible components of Igusa varieties for Shimura varieties of Hodge type, generalising results of Chai-Oort and Hida. Our strategy combines recent work of D’Addezio on monodromy of compatible local systems with a generalization of a method of Hida, and the Honda-Tate theory for Shimura varieties of Hodge type of Kisin - Madapusi Pera - Shin. As a corollary to our main theorem we deduce results about irreducible components of central leaves and Newton strata, in particular we provide a counterexample to the discrete Hecke-orbit conjecture of Chai-Oort.