# Constructing a derived zeta function

# Constructing a derived zeta function

The local zeta function of a variety $X$ over a finite field $k$ can be defined to be $\sum_{n \geq 1} |(\mathrm{Sym}^nX)(k)| t^n$. As this depends only on the point counts of symmetric powers of $X$ it is an invariant of the class of $X$ in the Grothendieck ring of varieties $K_0(\mathrm{Var}_k)$: the ring which is generated by varieties over $k$ modulo the relation that whenever $Z$ is a closed subvariety of $Y$ we have $[Y] = [Z] + [Y\backslash Z]$. In fact, the local zeta function can be thought of as having codomain in the big WItt ring. Both the Grothendieck ring of varieties and the Witt ring appear as the $0$-th $K$-theory groups of certain categories. In this talk we show how to lift the zeta function to $K$-theory to produce a map of spaces whose $\pi_0$ is the local zeta function and use this map to find interesting elements in higher $K$-groups corresponding to the Grothendieck ring of varieties.