Characterisations of flock quadrangles

We characterise the Hermitian and Kantor flock generalized quadrangles of order (q(2), q), q even, (associated with the linear and Fisher-Thas-Walker flocks of a quadratic cone, and the Desarguesian and Betten-Walker translation planes) in terms of a self-dual subquadrangle. Equivalently, we show that a herd which contains a translation oval must be associated with the linear or Fisher-Thas-Walker flock. This result is a consequence of the determination of all functions which satisfy a certain absolute trace equation whose form is remarkably similar to that of an equation arising in recent studies of ovoids in three-dimensional projective space of finite order q.