Counting points on Igusa varieties
de 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. The tools for the proof are, among other things, Fujiwara's trace formula, Honda-Tate theory and various Galois cohomology arguments concerning Hermitian modules, isocrystals, and conjugacy classes in algebraic groups. We fully stabilize the counting point formula in chapter 4 and apply the stable trace formula to give a description of the cohomology of Igusa varieties with nontrivial endoscopy, in the case of U (1, n -1) × U (0, n ) × ... × U (0, n ). The stabilization result is unconditional, but the description of the cohomology in the case of nontrivial endoscopy is the only conditional result in the current paper. In chapter 5 we compare our counting point formula and Arthur's L 2 -Lefschetz formula to prove a generalization of Harris-Taylor's second basic identity ([13, Thm V.5.4]). Combining this result with Mantovan's formula ([32, Thm 22]) and the knowledge of the cohomology of some Shimura varieties due to Kottwitz and Harris-Taylor, we obtain a formula for the cohomology of Rapoport-Zink spaces of EL-type. The last formula in particular implies Fargues's result ([8, Thm 8.1.4, 8.1.5]) on the supercuspidal part of the cohomology of Rapoport-Zink spaces.
