On Ovoids of Parabolic Quadrics

It is known that every ovoid of the parabolic quadric Q(4,q), q = ph, p prime, intersects every three-dimensional elliptic quadric in 1 mod p points. We present a new approach which gives us a second proof of this result and, in the case when p = 2, allows us to prove that every ovoid of Q(4,q) either intersects all the three-dimensional elliptic quadrics in 1 mod 4 points or intersects all the three-dimensional elliptic quadrics in 3 mod 4 points. We also prove that every ovoid of Q(4,q), q prime, is an elliptic quadric. This theorem has several applications, one of which is the non-existence of ovoids of Q(6,q), q prime, q > 3. We conclude with a 1 mod p result for ovoids of Q(6,q).