Symmetric spread sets

Some new results on symplectic translation planes are given using their representation by spread sets of symmetric matrices. We provide a general construction of symplectic planes of even order and then consider the special case of planes of order $q^2$ with kernel containing $\GF(q)$, stressing the role of Brown's theorem on ovoids containing a conic section. In particular we provide a criterion for a symplectic plane of even order $q^2$ with kernel containing $\GF(q)$ to be desarguesian. As a consequence we prove that a symplectic plane of even order $q^2$ with kernel containing $\GF(q)$ and admitting an affine homology of order $q-1$ or a Baer involution fixing a totally isotropic $2$-subspace is desarguesian. Finally a short proof that symplectic semifield planes of even order $q^2$ with kernel containing $\GF(q)$ are desarguesian is given.