Four dimensional symplectic geometry over the field with three elements and a moduli space of Abelian surfaces

Abstract

We study certain combinatorial structures related to the simple group of order 25920. Our viewpoint is to regard this group as $G = $ $P{\bf Sp} (4, F_3)$ and so we describe these configurations in terms of the symplectic geometry of the four dimensional space over the field with three elements. Because of the isogeny between $SO(5)$ and $Sp(4)$ we can also describe these in terms of an inner product space of dimension five over that same field. The study of these configurations goes back to the 19th-century, and we relate our work to that of previous authors. We also discuss a more modern connection: these configurations arise in the theory of the Igusa compactification of the moduli space of principally polarized Abelian surfaces with a level three structure.