Convergence of zeta functions for schemes of finite type over the integers

Solution 1:

Let's see. The key point is that the formula for the zeta function is that, if we consider the case of finite type over a finite field $X/k={\mathbb F}_q)$, there is an equivalence between the 'absolute' zeta function as you define it and the relative zeta function:

$$Z(X,k;t)=exp\{\sum_{m\leq 1} \frac{N_m}{m} t^m\},$$ where $N_m=card(X({\mathbb F}_{q^m}))$ is the number of rational points of $X$ over the unique extension of degree $m$ over the base field $k$.

The fact is that, if we have a rational point over an extension $k_m$, then the image is also defined over such extension! That's the rationale behind the bound provided (simple as hell!). That's why, if $x\in X(k_m)$, then so does $f(x)$, and there are clearly no more $k_m$-defined points on $X$ above $f(x)$ than $deg(f)$. And that's that!

In fact, the "absolute" zeta function is derived from the former by the substitution $t=q^{-s}.$

This is key, for in our case, to every finite morphism $f:X\to Y$ (in the case where $X, Y$ are defined over a finite field $k$) corresponds an easy bound $$N_m(X)\leq deg(f)\cdotp N_m(Y).$$ We do not think in terms of the field generated by the coordinates of our points, but we merely ask that these belong to a fixed field $k_m.$ This facilitates our count enormously, and enables us to use the degree of $f$ efficiently.

With this bound, you can obtain the desired convergence result by taking an Euler product over all (finite) characteristics.

I am pretty sure that Serre's paper contained this kind of background (don't have it here with me), but in any case Mircea Mustata has a lovely set of notes on the matter:

http://www.math.lsa.umich.edu/~mmustata/zeta_book.pdf

Needless to say, but I'll just remind that the dimension of an algebraic scheme is its Kronecker dimension, i.e. an elliptic curve over $\mathbb{Z}$ is of dimension $1+1=2$ (that's why it's called an arithmetic surface!). This does indeed count when you write bounds on the product, Euler-style.

Let us deal with the case where $X \to Spec(\mathbb{Z})$ misses a finite number of points of its target.

Taking logarithms, one sees that $\log \zeta(X_p,s)$ is equivalent to $C_p p^{-(s-d)}$, where $d$ is the fibre dimension of the structure map ($C_p$ is controlled essentially by $deg(f)$ and by $Y$, and is $\leq deg(f)$ if our $f$ has the affine space over $\mathbb{Z}$ as its target). It suffices to argue as in the case of the zeta function so as to establish that the infinite product converges for $Re(s-d)>1$, and since $\dim X=d+1$, we are done. I can imagine, though, that using the existence of a finite $f:X\to Y$ does imply, through the above bounds, that the absolute zeta function of $X$ converges whenever $Re(s)>\dim X$.

In the case where we have a finite morphism $f:X\to \mathbb{A}^n_{\mathbb{Z}},$ (or finite over an open subset of $Spec(\mathbb{Z}$) the zeta function of $Y$ corresponds to $\zeta(s-n)$, and the lower bound for $Re(s)$ is $n+1$, i.e. the Kronecker dimension of the schemes involved.

That's how I did it, way back when. Should you need further clarification, just ask.