Essential dimension via prismatic cohomology

Abstract: Let $f\colon Y→X$ 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 introduce a new technique, using 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 $p$.

commutative algebraalgebraic geometrynumber theory

Audience: researchers in the topic

Recent Advances in Modern p-Adic Geometry (RAMpAGe)

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