Essential Dimension via Prismatic Cohomology

Let f:Y→X be a finite covering map of complex algebraic varieties. The essential dimension of f is the smallest integer e such that, birationally, f arises as the pullback of a covering Y′→X′

 of dimension e, via a map X→X′. This invariant goes back to classical questions about reducing the number of parameters in a solution to a general nth degree polynomial, and appeared in work of Kronecker and Klein on solutions of the quintic. 


I will report on joint work with Benson Farb and Jesse Wolfson, where we use prismatic cohomology, to obtain lower bounds on the essential dimension of certain coverings. For example, we show that for an abelian variety A of dimension g the multiplication by p map A→A has essential dimension g for almost all primes p. Prismatic cohomology is used to obtain lower bounds on the coniveau filtration on mod p cohomology, analogous to that obtained in characteristic 0 using Hodge theory.



Mark Kisin


Harvard University