Counting Points on Igusa Varieties of Hodge Type

TL;DR

This paper extends the Langlands-Rapoport conjecture to Igusa varieties of Hodge type, providing a new point-counting formula that generalizes previous results from PEL type to Hodge type, advancing understanding in the Langlands program.

Contribution

It formulates and proves an analogue of the Langlands-Rapoport conjecture for Igusa varieties of Hodge type, generalizing Shin's PEL type point-counting formula using Kisin-Shin-Zhu techniques.

Findings

Proved an analogue of the Langlands-Rapoport conjecture for Hodge type Igusa varieties.

Derived a new point-counting formula for Hodge type Igusa varieties.

Generalized previous PEL type formulas to Hodge type.

Abstract

Igusa varieties are algebraic varieties that arise in the study of special fibers of Shimura varieties, and have demonstrated many applications in the Langlands program via a Langlands-Kottwitz style point-counting formula due to Shin in the case of PEL type. In this paper we formulate and prove an analogue of the Langlands-Rapoport conjecture for Igusa varieties of Hodge type, building off the work of Kisin in the case of Shimura varieties. We then use this description of the points to derive a point-counting formula for Igusa varieties of Hodge type, generalizing the fomula in PEL type of Shin, by drawing on the techniques of Kisin-Shin-Zhu.