In this short post we look at the general definition of a zeta function for a scheme of finite type over ${Spec {\mathbb Z}}$ and record its relationship with the zeta function of an algebraic variety ${X}$ over a finite field ${\mathbb{F}_q}$, defined as a kind of generating function for the number of points of ${X}$ over all finite extensions of ${\mathbb{F}_q}$.

Let ${X}$ be a scheme of finite type over ${Spec {\mathbb Z}}$. In fact, for concreteness, we start by looking at a single affine chart: imagining ${X=Spec A}$ as a geometrical space whose ring ${A}$ of functions is a quotient of some ring ${{\mathbb Z}[t_1,...,t_n]}$ of polynomials (in finitely many variables).

Then for every maximal ideal ${m}$ of ${A}$, the quotient ${A/m}$ is in fact a finite field, so we can define the norm ${N_m = |A/m|}$ to get a positive integer ${\geq 2}$ associated with ${m}$. We can use this to define the zeta function associated with ${X}$ to be the following formal Dirichlet series, by analogy with the classical Euler product formula.

$\displaystyle \zeta_X(s) = \prod_{m \text{ a maximal ideal of }A} \frac{1}{1-N_m^{-s}}.$

Notice that taking ${X=Spec {\mathbb Z}}$ we recover the classical Riemann zeta function, and taking the ring of integers of a number field recovers the zeta function of a number field. What happens if we take ${X}$ to be a variety over a finite field ${\mathbb{F}_q}$?

Well, ${log \zeta_X(s) = - \sum_{x} log(1-N_x^{-s})}$ where the sum is over all closed points ${x \in X}$.

What is a closed point of ${X}$? What is an ${\mathbb{F}_{q^r}}$-rational point of ${X}$? This is one of the confusing subtleties of scheme theory: these concepts are related but not exactly the same. Since the more classical definition of a zeta function is in terms of rational points, we will have to make this leap.

In fact, an ${\mathbb{F}_{q^r}}$-rational point (which we will also call a geometric point’, perhaps slightly nonstandardly) is simply a map ${Spec(\mathbb{F}_{q^r}) \rightarrow X}$. This makes sense for two reasons. Firstly, a point in ordinary geometry can be described as just a map from a one point space (or as a zero-simplex, if you like), so our definition seems plausible. Secondly, and more convincingly, in the case where ${X}$ is affine, for example the elliptic curve over ${{\mathbb Z}}$, ${y^2=x^3-x}$, such a map corresponds to a map of rings

$\displaystyle {\mathbb Z}[x,y]/(y^2-x^3+x) \rightarrow \mathbb{F}_{q^r}.$

In other words, it corresponds to a choice of values for ${x}$ and ${y}$ in ${\mathbb{F}_{q^r}}$ satisfying the governing equation – exactly what we should mean by specifying a ${\mathbb{F}_{q^r}}$-rational point.

On the other hand, a closed point is a maximal ideal of the ring of functions of an affine neighbourhood. Whenever we have a map ${Spec\mathbb{F}_{q^r} \rightarrow X}$, it corresponds to a map of rings whose image is a field. In other words, a map whose kernel is a maximal ideal. Therefore, every ${\mathbb{F}_{q^r}}$-rational point is associated with a unique closed point ${x}$. In fact, we get an induced map ${k(x) \rightarrow \mathbb{F}_{q^r}}$, where ${k(x)}$ is the residue field at ${x}$. Conversely, any such map can be extended and determines a geometric point.

So finally we have arrived at the relationship between closed points and geometric points. Namely, each closed point ${x}$ determines the a set of ${\mathbb{F}_{q^r}}$-rational points ${X_x}$ which can be canonically identified with ${Hom_{\mathbb{F}_q}(k(x), \mathbb{F}_{q^r})}$. But by basic Galois theory, this says that, if ${|k(x)|=q^d}$, ${X_x}$ is nonempty iff ${d|r}$, in which case ${|X_x|=d}$.

So let us return to our zeta function. Writing ${N_x=q^d}$, we can expand

$\displaystyle -log(1-N_x^{-s}) = \sum_{m \geq 1} \frac{q^{-sdm}}{m}.$

Now, set ${T=q^{-s}}$ and bearing in mind where we are trying to get (to an expression involving the numbers ${|X_x|}$), we can define ${N(m,x)}$ to be ${d}$ if ${d|m}$ and ${0}$ otherwise, and rewrite this sum as

$\displaystyle \sum_{m \geq 1} N(m,x)\frac{T^m}{m} = \sum_{m \geq 1} |Hom_{\mathbb{F}_q}(k(x), \mathbb{F}_{q^r})|\frac{T^m}{m}.$

Now, take the sum over all closed points ${x}$, and conclude that

$\displaystyle log \zeta_X(s) = \sum_{m \geq 1} (\sum_x |Hom_{\mathbb{F}_q}(k(x), \mathbb{F}_{q^r})|)\frac{T^m}{m} = \sum_{m \geq 1} |X(\mathbb{F}_{q^m})|\frac{T^m}{m}.$

This gives the familiar expression for the zeta function as an object whose logarithmic derivative is a generating function for the number of points of ${X}$ over all the finite field extensions of ${\mathbb{F}_q}$, as appears in the statement of the Weil Conjectures and is well studied. One can check further that given the analysis above, the Riemann Hypothesis’ for varieties of dimension ${d}$ over a finite field asserts that all the zeroes are on the lines ${Re(s) = 1/2,3/2,....,(2d-1)/2}$ and the poles are on the lines ${Re(s) = 0,1,2,...,d}$. Noting that ${{\mathbb Z}}$ is an object of dimension 1, the analogy with the classical Riemann hypothesis could not be clearer.