LOWER BOUNDS ON ESSENTIAL DIMENSION FOR CONGRUENCE COVERS OF MIXED SHIMURA VARIETIES

Abstract

We use the fixed point method to obtain 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 states 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 corresponding mixed Shimura datum. This generalizes theorems of Brosnan and Fakhruddin ({\cite[33, 34]{BrosnanFakhruddin}}) to the case of an arbitrary mixed Shimura datum. We also discuss a few examples of lower bounds in the last section.