Absolute zeta functions arising from ceiling and floor Puiseux polynomials

TL;DR

This paper introduces ceiling and floor Puiseux polynomials to characterize absolute zeta functions of schemes over finite fields, extending previous work and showing independence of isogeny class for elliptic curves.

Contribution

It defines new ceiling and floor Puiseux polynomials for schemes over $Q$, generalizing the concept of absolute zeta functions and providing a novel characterization.

Findings

Absolute zeta functions are characterized by ceiling and floor Puiseux polynomials.

For elliptic curves, the zeta functions are independent of isogeny class.

The approach extends previous interpolation methods for counting points over finite fields.

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.