ZETA FUNCTIONS. NO.9

YOSHIFUMI TSUCHIMOTO

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

An affine scheme X of finite type over ℤ, in turn, is related to a set f_{1}, … f_{m} of polynomial equations of coefficients in ℤ, and is written as Spec(A) for a ring A = ℤ[X_{1}, … X_{n}]/(f_{1}, … f_{m}) of finite type over ℤ, We consider quasi coherent sheaves over these objects. When X is affine (X = 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 ℤ, we put

ζ(X, s) = ∏_{x∈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(∑_{m=1}^{∞} \frac{#X(𝔽_{q^{m}})}{m}t^{m}).

Proposition 9.1. Let X be a scheme of finite type over 𝔽_{q}. Then we have

ζ(X, s) = Z(X, q^{−s}).

YOSHIFUMI TSUCHIMOTO

Proof.

log(ζ(X, s))
= − ∑_{𝔪∈Spm(X)} log(1 − N(𝔪)^{−s})
= ∑_{𝔪} ∑_{r=1}^{∞} \frac{N(𝔪)^{−rs}}{r}
= ∑_{u=1}^{∞} ∑_{\begin{matrix}𝔪 \\ [R/𝔪:𝔽_{q}]=u \end{matrix}} ∑_{r=1}^{∞} \frac{N(𝔪)^{−rs}}{r}
= ∑_{u=1}^{∞} \frac{#X(𝔽_{q^{u}})_{∗}}{u} ∑_{r=1}^{∞} \frac{q^{−urs}}{r} = (♡).

where we put

X(𝔽_{q^{u}})_{∗} = X(𝔽_{q^{u}}) \ ∪_{t<u}(X(𝔽_{q^{t}})).

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

(♡) = ∑_{u=1}^{∞} ∑_{r=1}^{∞} \frac{#X(𝔽_{q^{u}})_{∗}}{ur} t^{ur}

= ∑_{m} \frac{t^{m}}{m} ∑_{ur=m} #X(𝔽_{q^{u}})_{∗}

= ∑_{m} \frac{t^{m}}{m} #X(𝔽_{q^{m}})

= − log(Z(X, t))

□

Proposition 9.2. Let X be a scheme of finite type over ℤ. then we have

ζ(X, s) = ∏_{p:prime} ζ((X mod p), s).

Where we define X mod p as a fiber product X ×_{Spec ℤ} Spec 𝔽_{p}.

Lemma 9.3.

Z(X × 𝔸^{1}, t) = Z(X, qt).
ζ(X × 𝔸^{1}, s) = ζ(X, s − 1)