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

### Najmuddin Fakhruddin

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India### Rijul Saini

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India

## 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.

## Cite this article

Najmuddin Fakhruddin, Rijul Saini, Finite groups scheme actions and incompressibility of Galois covers: beyond the ordinary case. Doc. Math. 27 (2022), pp. 151–182

DOI 10.4171/DM/868