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

Abstract

Inspired by recent work of Farb, Kisin and Wolfson [8], 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.

Essential dimension via prismatic cohomology

“…In particular, it allows us to give lower bounds on ed(Y /X) in many cases when X is proper. Previously, lower bounds on the essential dimension of coverings of proper varieties were known only in very special cases [CT02,FKW19,FS21]. In fact our results apply to the so called p-essential dimension ed(Y /X; p) (p a prime), where one is allowed to pull back the covering not just to Zariski opens, but to auxiliary coverings of prime to p degree [RY00].…”

“…The conditions on G and p in Theorem 3 are completely different from those considered in [FKW19,FS21]. For example, in many cases when the results of loc.…”

“…When X is an abelian variety, this confirms (almost all of) a conjecture of Brosnan [FS21, Conj. 6.1], which was previously known only for either a very general abelian variety, or in dimension at most 3 for a positive density set of primes [FS21]. The result for a product of curves was known only for a generic product of elliptic curves, by a result of Gabber [CT02].…”

The behavior of essential dimension under specialization

“…The assumption that A and R are complete may be dropped; see Theorem 6.4. In the case, where G is a finite group and p = 0, the inequality ed k 0 (α K 0 ) ed k (α K ) was noted in [FS21,Remark 6.3]. A version of Theorem 1.2 for essential dimension at a prime will be proved in the Appendix; see Theorem A.1.…”

The behavior of essential dimension under specialization II

“…Fakhruddin and R. Saini [FS21]. Some arise as congruence covers of Shimura varieties, others from actions of subgroups of X[p] ≃ (Z/pZ) 2n on a complex abelian variety X (not necessarily a product of elliptic curves).…”