D-elliptic sheaves and odd Jacobians

We examine the existence of rational divisors on modular curves of D-elliptic sheaves and on Atkin–Lehner quotients of these curves over local fields. Using a criterion of Poonen and Stoll, we show that in infinitely many cases the Tate–Shafarevich groups of the Jacobians of these Atkin–Lehner quotients have non-square orders.