On m-ovoids of W3(q)

We show that the generalized quadrangle W3(q) for odd q has exponentially many -ovoids, thus implying that the generalized quadrangle Q(4,q) has exponentially many hemisystems for odd q. For q even, we show that W3(q) has m-ovoids for all integers m, 1⩽m⩽q. Stabilizers are determined, and some computer results are given.