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| Auteurs principaux: | , |
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| Format: | Preprint |
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2023
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| Accès en ligne: | https://arxiv.org/abs/2308.03232 |
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| _version_ | 1866910450184093696 |
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| author | Hirakawa, Yoshinosuke Tomita, Takuki |
| author_facet | Hirakawa, Yoshinosuke Tomita, Takuki |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_03232 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Absolute zeta functions arising from ceiling and floor Puiseux polynomials Hirakawa, Yoshinosuke Tomita, Takuki Number Theory 14G10 (Primary) 11M41, 11R59 (Secondary) 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. |
| title | Absolute zeta functions arising from ceiling and floor Puiseux polynomials |
| topic | Number Theory 14G10 (Primary) 11M41, 11R59 (Secondary) |
| url | https://arxiv.org/abs/2308.03232 |