$E_{\infty }$-Ring structures on the $K$-theory of assemblers and point counting

Abstract

We construct a monoidal structure on the category of assemblers. As an application of this, we construct a derived local zeta-function which takes a variety over a finite field to the set of points over the separable closure, and use the structure of this map to detect interesting elements in $K_1({\mathbf{Var}}_k)$.