Rationality of $\mbox{dlog}$ $\mathbb{A}^1$-zeta functions

Abstract

For every smooth proper scheme over a finite field $\mathbb{F}_q$, Bilu, Ho, Srinivasan, Vogt, and Wickelgren introduced the dlog zeta function with coefficients in the Grothendieck-Witt ring $\mathrm{GW}(\mathbb{F}_q)$, enriching the dlog of the classical Weil zeta function with coefficients in $\mathbb{Z}$. They defined a notion of dlog rationality of such dlog zeta functions, which enriches the rationality of the Weil zeta function, and showed the dlog rationality for simple cellular schemes. In this paper, we show that for any smooth proper schemes over $\mathbb{F}_q$, the dlog zeta function is rational, but not necessarily dlog rational.