Irrationality of motivic zeta functions

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.