Local diophantine properties of modular curves of $\mathcal{D}$-elliptic sheaves

We study the existence of rational points on modular curves of $\mathcal{D}$-elliptic sheaves over local fields and the structure of special fibres of these curves. We discuss some applications which include finding presentations for arithmetic groups arising from quaternion algebras, finding the equations of modular curves of $\mathcal{D}$-elliptic sheaves, and constructing curves violating the Hasse principle.