Moduli of Abelian surfaces with a $(1,p^2)$ polarisation

The moduli space of abelian surfaces with a polarisation of type $(1,p^2)$ for $p$ a prime was studied by O'Grady in [7], where it is shown that a compactification of this moduli space is of general type if $p\geqslant 17$. We shall show that in fact this is true if $p\geqslant 11$. Our methods overlap with those of [7], but are in some important ways different. We borrow notation freely from that paper when discussing the geometry of the moduli space.