Mathematics – Number Theory
Scientific paper
2010-10-08
Mathematics
Number Theory
4 pages
Scientific paper
For a scheme $X$ whose $\mathbb F_q$-rational points are counted by a polynomial $N(q)=\sum a_iq^i$, 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 $\zeta(r+N-s) = (-1)^\chi ( \zeta(s) )^{(-1)^r}$.
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