On degenerations of D-shtukas

Abstract

Let $F$ be a global field of positive characteristic, and let $D$ be a maximal order in a central division algebra over $F$. We give a precise properness criterion for moduli spaces of $D$-shtukas with general modifications, which includes the statement that for some division algebras, the space of ordinary $D$-shtukas is not proper. The proof is based on a result on the smoothness of a suitable stack of complete homomorphisms of $D$-modules of rank $1$