Lower Bounds on Essential Dimension for Congruence Covers of Mixed Shimura Varieties

We use the fixed point method and toroidal compactifications to establish general lower bounds for the essential dimension of congruence covers $\Gamma' \backslash \mathcal{X}^0 \rightarrow \Gamma \backslash \mathcal{X}^0$ of mixed Shimura varieties. Our main result shows that $\mathrm{ed}_{\mathbb{C}}(\Gamma' \backslash \mathcal{X}^0 \rightarrow \Gamma \backslash \mathcal{X}^0; p)$ is bounded from below by the dimension of certain unipotent subgroups associated with the rational boundary components of the given mixed Shimura datum. This generalizes theorems of Brosnan and Fakhruddin to the case of an arbitrary mixed Shimura datum. As a consequence, we obtain incompressibility results for congruence covers of universal families of principally polarized abelian varieties. We also describe explicit fixed points for $(GL_2, \mathcal{H}_2)$ and $(V \rtimes GL_2, \mathcal{Y}_2)$.