In his lecture "Zeta functions and $L$-functions", Serre presents a very elegant proof of the convergence of the zeta function $ \zeta (X,s) = \prod_{x \in |X|} (1- N(x)^{-s})^{-1}$ in the half plane $R(s) > dim(X)$, where $X$ is a scheme of finite type over $\mathbb{Z}$, $|X|$ the set of closed points of $X$ and $N(x)$ the number of elements in the residue field $k(x)$.
He reduces the claim to the case where $X = Spec \, A[x_1, \ldots x_n]$ and $A$ is either $\mathbb{Z}$ or $\mathbb{F}_p$.
The decisive input is the following lemma:
a) If $X$ is the finite union of the schemes $X_i$, and the claim holds for all $X_i$, then it holds for $X$.
b) If $f: X \to Y$ is finite and the claim holds for $Y$, then it holds for $X$ as well.
I've been trying to prove b) but I seem to be missing something. Here's what I've tried so far: I was considering $\zeta(X,s) = \prod_{y \in |Y|} \zeta(X_y \, ,s)$, where $X_y$ is the fiber of $f$ at $y$. I now the fibers are finite but I don't know how to connect this with the fact that $\zeta(Y,s)$ converges. Is it true that the residue field $k(y)$ is a finite extension of $k(x)$ for all $x \in X_y$ (of degree $\deg f$)? I know this is the case for the function fields.
Any help is very appreciated!
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.