Point Counting on Igusa Varieties for function fields

TL;DR

This paper advances the understanding of Igusa varieties over moduli stacks of global G-shtukas, providing new calculations of Hecke actions and insights into local G-shtukas, with implications for the Langlands program.

Contribution

It introduces novel results on local G-shtukas in both equal and unequal characteristic and applies these to Barsotti-Tate groups and Shimura varieties.

Findings

Calculated Hecke action on cohomology of Igusa varieties

Proved new results on local G-shtukas in various characteristics

Discussed applications to Barsotti-Tate groups and Shimura varieties

Abstract

Igusa varieties over the special fibre of Shimura varieties have demonstrated many applications to the Langlands program via Mantovan's formula and Shin's point counting method. In this paper we study Igusa varieties over the moduli stack of global $\Gscr$-shtukas and (under certain conditions) calculate the Hecke action on its cohomology. As part of their construction we prove novel results about local $G$-shtukas in both equal and unequal characteristic and also discuss application of these results to Barsotti-Tate groups and Shimura varieties.