Applications of number theory to ovoids and translation planes (Q1818589)

A translation plane of order \(p^2\) is said to be of Mason or of Mason-Ostrom type if its automorphism group contains the group \(\text{SL}_2(9)\) or an extension of an extraspecial group of order \(32\) by the symmetric group \(\text{S}_5\), respectively. Examples have been known for small values of the prime \(p\). In the present interesting paper, the authors prove the following Theorem. For each prime \(p\geq 11\) there exist translation planes of order \(p^2\) of Mason type and of Mason-Ostrom type. The proof uses the fact that the Klein correspondence relates spreads in \({\mathbb F}_p^{ 4}\) to \(6\)-dimensional ovoids [cf. \textit{G. Mason} and \textit{E. Shult}, Geom. Dedicata 21, 29--50 (1986; Zbl 0593.51006)]. These ovoids are obtained as projections from \(8\)-dimensional binary ovoids of \textit{J. H. Conway} et al. [Geom. Dedicata 26, No. 2, 157--170 (1988; Zbl 0643.51015)]. This requires a detailed analysis of specific quadratic forms.