Ovoids of the quadric Q(2n, q)

We consider ovoids of the non-singular quadric Q(2n, q) in PG(2n, q). It is known that Q(6, q) with q = 2h has no ovoid, while Q(6, q) with q = 3h admits ovoids. Here we prove that if q is odd, q ≠ 3, and every ovoid of the non-singular quadric Q(4, q) in PG(4, q) is an elliptic quadric, then Q(6, q), and hence also Q(2n, q) with n ⩾ 3, has no ovoid. As a corollary, it follows that Q(2n, 5) and Q(2n, 7), n ⩾ 3, have no ovoid.