Irrationality of motivic zeta functions

TL;DR

The paper demonstrates that the motivic zeta function of a certain K3 surface over the rationals is not rational, providing a counterexample to a previous conjecture by Denef and Loeser.

Contribution

It constructs a specific K3 surface over $Q$ whose motivic zeta function is irrational, disproving the conjecture of rationality.

Findings

Existence of a K3 surface with irrational motivic zeta function

Disproof of the conjecture that motivic zeta functions are always rational

Counterexample in the Grothendieck ring context

Abstract

Let $K_0(\mathrm{Var}_{\mathbb{Q}})[1/\mathbb{L}]$ denote the Grothendieck ring of $\mathbb{Q}$-varieties with the Lefschetz class inverted. We show that there exists a K3 surface X over $\mathbb{Q}$ such that the motivic zeta function $\zeta_X(t) := \sum_n [\mathrm{Sym}^n X]t^n$ regarded as an element in $K_0(\mathrm{Var}_{\mathbb{Q}})[1/\mathbb{L}][[t]]$ is not a rational function in $t$, thus disproving a conjecture of Denef and Loeser.