Moduli spaces of local G-shtukas

TL;DR

This paper reviews the theory of local G-shtukas and their moduli spaces, highlighting their connections to global G-shtukas, affine Deligne-Lusztig varieties, and period maps, emphasizing recent developments and ongoing research.

Contribution

It provides a comprehensive overview of local G-shtukas, their moduli spaces, and their relations to other key objects in arithmetic geometry, consolidating recent advances.

Findings

Overview of local G-shtukas and their moduli spaces

Connections between local and global G-shtukas

Properties of special and generic fibers, including affine Deligne-Lusztig varieties

Abstract

We give an overview of the theory of local G-shtukas and their moduli spaces that were introduced in joint work of U.~Hartl and the author, and in the past years studied by many people. We also discuss relations to moduli of global G-shtukas, properties of their special fiber through affine Deligne-Lusztig varieties and of their generic fiber, such as the period map.