On properness of moduli stacks of $D^{\times}$-shtukas over ramified legs

TL;DR

This paper establishes explicit conditions for the properness of moduli stacks of $D^{ imes}$-shtukas over ramified points, refining previous results and analyzing stratification properties in the context of algebraic geometry over function fields.

Contribution

It provides a new explicit criterion for properness of these moduli stacks, extending Lau's work to ramified cases and examining stratification non-emptiness.

Findings

Derived explicit properness conditions based on local invariants.

Refined Lau's properness results to include ramified loci.

Proved non-emptiness of Newton and Kottwitz--Rapoport strata.

Abstract

Given a maximal order $D$ of a central division algebra over a global function field $F$, we prove an explicit sufficient condition for moduli stacks of $D^\times$-shtukas to be proper over a finite field in terms of the local invariants of $D$ and bounds. Our proof is a refinement of E.~Lau's result (Duke Math. J. 140 (2007)), which showed the properness of the leg morphism (or characteristic morphism) away from the ramification locus of $D$. We also establish non-emptiness of Newton and Kottwitz--Rapoport strata for moduli stacks of $B^\times$-shtukas, where $B$ is a maximal order of a central simple algebra over $F$.