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

TL;DR

This paper introduces a method leveraging finite group scheme actions over mixed characteristic DVRs to establish lower bounds on the essential dimension of covers, demonstrating p-incompressibility in certain Shimura varieties and advancing conjectures on abelian varieties.

Contribution

It develops a new approach using finite group scheme actions to analyze essential dimension and applies it to prove p-incompressibility of specific Shimura covers and abelian variety maps.

Findings

Proves p-incompressibility for certain unitary Shimura varieties.

Establishes lower bounds for essential dimension using group scheme actions.

Progresses towards Brosnan's conjecture on p-incompressibility of multiplication by p maps.

Abstract

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.