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arXiv · 1802.03661

Irrationality of motivic zeta functions

Michael Larsen·Valery Lunts
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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 $ζ_X(t) := \sum_n [\mathrm{Sym}^n X]t^n$ 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.

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BibTeXRIS

Michael Larsen, Valery Lunts. 2018-09-06. Irrationality of motivic zeta functions. https://doi.org/10.1215/00127094-2019-0035

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