Point Counting on Igusa Varieties for function fields
Paul Hamacher, Wansu Kim
TL;DR
The paper constructs Igusa varieties over moduli stacks of global G-shtukas in global function fields and proves a Shin-type point-counting formula, yielding a cohomological description in the division-algebra case. It develops a comprehensive local–global framework with Igusa covers for local G-shtukas, Beilinson–Drinfeld bounds, and adelic descent, enabling explicit Hecke actions and fixed-point counts. A full trace formula is established in the division-algebra setting (n<p), expressing cohomological traces as sums over automorphic representations via Red^b and Jacquet–Langlands transfers, together with a robust elliptic-term analysis and Kottwitz–Igusa triples. The work also demonstrates applications to Barsotti–Tate groups, Shimura varieties, and bounds/purity phenomena, and outlines conjectural extensions to the general function-field Igusa setting beyond division algebras. Overall, it builds a function-field analogue of the classical Igusa–Rapoport framework, providing tools for Langlands-type correspondences through geometric and automorphic inputs and suggesting future directions via Lefschetz-type formulas and elliptic terms beyond the division-algebra case.
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.