Absolute zeta functions arising from ceiling and floor Puiseux polynomials

Abstract

For the [Formula: see text]-lift [Formula: see text] of a monoid scheme [Formula: see text] of finite type, Deitmar et al. calculated its absolute zeta function by interpolating [Formula: see text] for all prime powers [Formula: see text] using the Fourier expansion. This absolute zeta function coincides with the absolute zeta function of a certain polynomial. In this paper, we characterize the polynomial as a ceiling polynomial of the sequence [Formula: see text], which we introduce independently. Extending this idea, we introduce a certain pair of absolute zeta functions of a separated scheme [Formula: see text] of finite type over [Formula: see text] by means of a pair of Puiseux polynomials which estimate “[Formula: see text]” for sufficiently large [Formula: see text]. We call them the ceiling and floor Puiseux polynomials of [Formula: see text]. In particular, if [Formula: see text] is an elliptic curve, then our absolute zeta functions of [Formula: see text] do not depend on its isogeny class.