The Siegel modular variety $A_2(3)$ which parametrizes abelian surfaces with split level $3$ structure is birational to the Burkhardt quartic threefold. This was shown to be rational over $\mathbb{Q}$ by Bruin and Nasserden. What can we say about its twist $A_2(\rho)$ for a Galois representation \rho valued in $\operatorname{GSp}(4, \mathbb{F}_3)$? While it is not rational in general, it is unirational over $\mathbb{Q}$ by a map of degree at most $6$. In joint work with Frank Calegari and David Roberts, we obtain an explicit description of the universal object over a degree $6$ cover using invariant theoretic ideas. Similar ideas work in other cases, and hence for $(g,p) = (1,2), (1,3), (1,5), (2,2), (2,3)$ and $(3,2)$, any Galois representation $\rho$ valued in $\operatorname{GSp}(2g,\mathbb{F}_p)$ with cyclotomic similitude character arises from the $p$-torsion of a $g$-dimensional abelian variety. When $(g,p)$ is not one of these six tuples, we discuss a local obstruction for representations to arise as torsion.
