Absolute zeta functions arising from ceiling and floor Puiseux polynomials

Abstract

For the $\mathbb{Z}$-lift $X_\mathbb{Z}$ of a monoid scheme $X$ of finite type, Deitmar-Koyama-Kurokawa calculated its absolute zeta function by interpolating $\#X_\mathbb{Z}(\mathbb{F}_q)$ for all prime powers $q$ using the Fourier expansion. This absolute zeta function coincides with the absolute zeta function of a certain polynomial. In this article, we characterize the polynomial as a ceiling polynomial of the sequence $\left(\#X_\mathbb{Z}(\mathbb{F}_q)\right)_q$, which we introduce independently. Extending this idea, we introduce a certain pair of absolute zeta functions of a separated scheme $X$ of finite type over $\mathbb{Q}$ by means of a pair of Puiseux polynomials which estimate "$\#X(\mathbb{F}_{p^m})$" for sufficiently large $p$. We call them the ceiling and floor Puiseux polynomials of $X$. In particular, if $X$ is an elliptic curve, then our absolute zeta functions of $X$ do not depend on its isogeny class.