Finite groups scheme actions and incompressibility of Galois covers: beyond the ordinary case

Abstract

Inspired by recent work of B. Farb et al. [Compos. Math. 157, No. 11, 2407–2432 (2021; Zbl 1490.14042)], we develop a method for using actions of finite group schemes over a mixed characteristic $\mathrm{dvr}\mathbb{R}$ to get lower bounds for the essential dimension of a cover of a variety over $K=\text{Frac}(\mathbb{R})$. We then apply this to prove $p$-incompressibility for congruence covers of a class of unitary Shimura varieties for primes $p$ at which the reduction of the Shimura variety (at any prime of the reflex field over $p)$ does not have any ordinary points. We also make some progress towards a conjecture of Brosnan on the $p$-incompressibility of the multiplication by $p$ map of an abelian variety.