Counting points on Igusa varieties

Sug Woo Shin

Igusa varieties are smooth varieties over [Special characters omitted.] and higher-dimensional analogues of Igusa curves, which were used to study the bad reduction of modular curves ([15]). Igusa varieties played a crucial role in the work of Harris and Taylor on the Langlands correspondence ([13]). Later the notion of Igusa varieties was generalized by Mantovan ([31], [32]). This paper is an attempt to understand the cohomology of Shimura varieties, Igusa varieties and Rapoport-Zink spaces in connection with the Langlands correspondence, with emphasis on the role of Igusa varieties. Our paper is organized in five chapters. The first two chapters contain preliminary materials as well as the definitions and basic properties of Shimura varieties, Igusa varieties and Rapoport-Zink spaces. Our first main result is the counting point formula for Igusa varieties at the end of chapter 3....