Functional equations for zeta functions of ${\mathbb{F}}_1$-schemes

Abstract

For a scheme $X$ whose ${\mathbb{F}}_q$-rational points are counted by a polynomial $N(q)=\sum a_i{q}^i$, the ${\mathbb{F}}_1$-zeta function is defined as $\zeta (s)=\prod (s-i{)}^{-a_i}$. Define $\chi =N(1)$. In this paper we show that if $X$ is a smooth projective scheme, then its ${\mathbb{F}}_1$-zeta function satisfies the functional equation $\zeta (n-s)=(-1{)}^{\chi }\zeta (s)$. We further show that the ${\mathbb{F}}_1$-zeta function $\zeta (s)$ of a split reductive group scheme $G$ of rank $r$ with $N$ positive roots satisfies the functional equation $\zeta (r+N-s)=(-1{)}^{\chi }(\zeta (s){)}^{(-1{)}^r}$.

4 pages

References in corpus (3)

• Mapping F_1-land:An overview of geometries over the field with one element
• Algebraic groups over the field with one element
• On the notion of geometry over $\F_1$