Elation generalized quadrangles of order (q, q2) q even, with a classical subquadrangle of order q

One of the famous open problems in Finite Geometry is the classification of ovoids in a projective 3-space PG(3, q) over the finite field with q elements, q even. For q odd, this classification was obtained in 1955 by A. Barlotti [1] and G. Panella [10]. A breakthrough result was a recent one of M. R. Brown, who showed in [3] that when such an ovoid has a conic plane section, the ovoid must be an elliptic quadric. Even more recently, M. R. Brown and M. Lavrauw [4] generalized this theorem by obtaining a similar result for higher dimensions. Both results have equivalent statements in the theory of (translation) generalized quadrangles. In this note, we improve these results by showing that an elation generalized quadrangle of order (q, q(2)), q even, with a Q(4,q) subGQ containing the elation point arises from a non-singular elliptic quadric in PG(5, q). The theorem itself arises as a corollary of a more general observation which works for all characteristics. There is a wealth of consequences.