Monodromy and irreducibility of Igusa varieties

Abstract: Igusa varieties are smooth varieties in characteristic $p$ arising naturally as covers of certain subvarieties (central leaves) of Shimura varieties, for example of the ordinary locus of the modular curve. The $\ell$-adic cohomology of Igusa varieties acts as a bridge between the cohomology of Rapoport-Zink spaces (local) and the cohomology of Shimura varieties (global), and it is therefore very interesting to study this cohomology. In this talk I will discuss recent joint work with Luciena Xiao Xiao, where we compute the 0th cohomology group. This is equivalent to determining the irreducible components of Igusa varieties, and our results generalise results of Hida and Chai-Oort. Our strategy combines recent work of D’Addezio on monodromy of compatible local systems with a generalisation of a method of Hida, and the Honda-Tate theory for Shimura varieties of Hodge type of Kisin--Madapusi Pera--Shin.

number theory

Audience: researchers in the topic

UCLA Number Theory Seminar