Essential dimension via prismatic cohomology

Abstract: Let $f:Y \rightarrow X$ 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 $Y'\rightarrow X'$ of dimension $e$, via a map $X \rightarrow X'$. This invariant goes back to classical questions about reducing the number of parameters in a solution to a general $n$-th 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 $A \rightarrow A$ has essential dimension $g$ for almost all primes $p$.

algebraic geometrynumber theory

Audience: researchers in the topic

Comments: Meeting ID: 908 611 6889 , Passcode: order of the symmetric group on 9 letters (type the 6-digit number)

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FGC-HRI-IPM Number Theory Webinars

Series comments: password is 848084

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