Zeta functions. No.9

Yoshifumi Tsuchimoto

In this lecture we make use of a scheme $X$ of finite type over $\mathbb{Z}$ , It is a patchwork of affine schemes of finite type over $\mathbb{Z}$ ,

An affine scheme $X$ of finite type over $\mathbb{Z}$ , in turn, is related to a set $f_1,\dots f_m$ of polynomial equations of coefficients in $\mathbb{Z}$ , and is written as $\operatorname{Spec}(A)$ for a ring $A=\mathbb{Z}[X_1,\dots X_n]/(f_1,\dots f_m)$ of finite type over $\mathbb{Z}$ , We consider quasi coherent sheaves over these objects. When $X$ is affine ( $X=\operatorname{Spec}(A)$ ), the category of quasi coherent sheaves over $X$ is equivalent to the category of $A$ -modules.

For any scheme $X$ of finite type over $\mathbb{Z}$ , we put

$$\zeta(X,s)=\prod_{x \in \operatorname{Spm}(X)} (1-(Nx)^{-s})^{-1}$$

It is equal to the zeta function of the category of quasi coherent sheaves on $X$ .

Recall we have defined the congruent zeta function as

$$Z(X,t)=\exp( \sum_{m=1}^\infty \frac{\char93 X(\mathbb{F}_{q^m}) }{m}t^m).$$

PROPOSITION 9.1   Let $X$ be a scheme of finite type over $\mathbb{F}_q$ . Then we have

$$\zeta(X,s)=Z(X,q^{-s}).$$

PROOF..

$$\log(\zeta(X,s))$$

$$= -\sum_{\mathfrak{m} \in \operatorname{Spm}(X)} \log(1-N(\mathfrak{m})^{-s})$$

$$= \sum_{\mathfrak{m}} \sum_{r=1}^\infty \frac{N(\mathfrak{m})^{-rs} }{r}$$

$$= \sum_{u=1}^\infty \sum_ {\substack{\m... ...hfrak{m} : \mathbb{F}_q]=u}} \sum_{r=1}^\infty \frac{N(\mathfrak{m})^{-rs} }{r}$$

$$= \sum_{u=1}^\infty \frac{\char93 X(\mathbb{F}_{q^u})_*}{u} \sum_{r=1}^\infty \frac{q^{-urs}}{r}=(\heartsuit).$$

where we put

$$X(\mathbb{F}_{q^u})_*= X(\mathbb{F}_{q^u}) \setminus \cup _{t<u}(X(\mathbb{F}_{q^t})).$$

Now let us put $t=q^{-s}$ and proceed further.

$$(\heartsuit) = \sum_{u=1}^\infty \sum_{r=1}^\infty \frac{\char93 X(\mathbb{F}_{q^u})_*}{ur} t^{ur}$$

$$=\sum_m \frac{t^m}{m} \sum_{ur=m} \char93 X(\mathbb{F}_{q^u})_*$$

$$= \sum_m \frac{t^m}{m}\char93 X(\mathbb{F}_{q^m})$$

$$=-\log(Z(X,t))$$

$\qedsymbol$

PROPOSITION 9.2   Let $X$ be a scheme of finite type over $\mathbb{Z}$ . then we have

$$\zeta(X,s)=\prod_{p:{\rm { prime}}}\zeta((X \mod p),s).$$

Where we define $X\mod p$ as a fiber product $X \times_{\operatorname{Spec}\mathbb{Z}} {\operatorname{Spec}\mathbb{F}_p}$ .

LEMMA 9.3  

$$\zeta(X\times \mathbb{A}^1,s)=\zeta(X,s-1)$$