FINITE GROUPS SCHEME ACTIONS AND INCOMPRESSIBILITY OF GALOIS COVERS: BEYOND THE ORDINARY CASE

Inspired by recent work of Farb, Kisin and Wolfson, we develop a method for using actions of finite group schemes over a mixed characteristic dvr R to get lower bounds for the essential dimension of a cover of a variety over K = Frac(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.

Automated summary

Inspired by recent work, this paper develops a method for obtaining lower bounds for the essential dimension of a cover of a variety using actions of finite group schemes. The method is then applied to prove p-incompressibility for congruence covers of a class of unitary Shimura varieties and make progress towards a conjecture on the p-incompressibility of the multiplication by p map of an abelian variety.