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

TL;DR

This paper develops a monoidal structure on assemblers and introduces a derived local zeta-function to analyze the $K$-theory of varieties over finite fields, revealing new elements in $K_1( ext{Var}_k)$.

Contribution

It constructs an $E_$-ring structure on the $K$-theory of assemblers and defines a derived local zeta-function for varieties over finite fields.

Findings

Constructed a monoidal structure on the category of assemblers.

Defined a derived local zeta-function mapping varieties to point sets.

Used the structure to identify interesting elements in $K_1( ext{Var}_k)$.

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 $K_1(\mathbf{Var}_k)$.