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

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 $F$.