$\mathcal{D}$-elliptic sheaves and the Hasse principle

Let $p$ be a rational prime, $q>1$ a power of $p$ and $F={\mathbb{F}}_q(t)$. For an integer $d\ge 2$, let $D$ be a central division algebra over $F$ of dimension $d^2$ which is split at $\mathrm{\infty }$ and has invariant ${\mathrm{i}\mathrm{n}\mathrm{v}}_x(D)=1\mathrm{/}d$ at any place $x$ of $F$ at which $D$ ramifies. Let $X^D$ be the Drinfeld–Stuhler variety, the coarse moduli scheme of the algebraic stack over $F$ classifying $\mathcal{D}$-elliptic sheaves. In this paper, we establish various arithmetic properties of $\mathcal{D}$-elliptic sheaves to give an explicit criterion for the non-existence of rational points of $X^D$ over a finite extension of $F$ of degree $d$. As an application, for $d=2$, we present explicit infinite families of quadratic extensions of $F$ over which the curve $X^D$ violates the Hasse principle.