Essential dimension via prismatic cohomology

Mark Kisin (Harvard)

14-Oct-2021, 19:00-20:00 (2 years ago)

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 $n^{\rm th}$ 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

( paper )

Comments: Note the unusual time and place: Thursday at 3pm in 2-449.


MIT number theory seminar

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Organizers: Edgar Costa*, Siyan Daniel Li-Huerta*, Bjorn Poonen*, David Roe*, Andrew Sutherland*, Robin Zhang*, Wei Zhang*, Shiva Chidambaram*
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