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.

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