Motivic classes of classifying stacks of algebraic groups

Abstract: The Grothendieck ring of algebraic stacks was introduced by Ekedahl in 2009. It may be viewed as a localization of the more classical Grothendieck ring of varieties. If $G$ is a finite group, then the class $\{BG\}$ of its classifying stack $BG$ is equal to 1 in many cases, but there are examples for which $\{BG\}\neq 1.$ When $G$ is connected, $\{BG\}$ has been computed in many cases in a long series of papers, and it always turned out that $\{BG\}*\{G\}=1.$ We exhibit the first example of a connected group $G$ for which $\{BG\}*\{G\}\neq 1.$ As a consequence, we produce an infinite family of non-constant finite \'etale group schemes $A$ such that $\{BA\}\neq 1.$

number theory

Audience: researchers in the topic

Columbia CUNY NYU number theory seminar