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

TL;DR

This paper establishes functional equations for the $_1$-zeta functions of smooth projective schemes and split reductive group schemes, revealing symmetry properties and extending the understanding of $_1$-zeta functions in algebraic geometry.

Contribution

It proves the functional equations for $_1$-zeta functions of certain schemes, linking their properties to geometric and group-theoretic structures.

Findings

$_1$-zeta function of smooth projective schemes satisfies a specific functional equation.

$_1$-zeta function of split reductive group schemes satisfies a generalized functional equation.

The functional equations involve the rank, roots, and Euler characteristic of the schemes.

Abstract

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 $\zeta(r+N-s) = (-1)^\chi ( \zeta(s) )^{(-1)^r}$.