In this short post we look at the general definition of a zeta function for a scheme of finite type over and record its relationship with the zeta function of an algebraic variety over a finite field , defined as a kind of generating function for the number of points of over all finite extensions of .

Let be a scheme of finite type over . In fact, for concreteness, we start by looking at a single affine chart: imagining as a geometrical space whose ring of functions is a quotient of some ring of polynomials (in finitely many variables).

Then for every maximal ideal of , the quotient is in fact a finite field, so we can define the norm to get a positive integer associated with . We can use this to define the *zeta function* associated with to be the following formal Dirichlet series, by analogy with the classical Euler product formula.

Notice that taking 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 to be a variety over a finite field ?

Well, where the sum is over all closed points .

What is a closed point of ? What is an -rational point of ? 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 -rational point (which we will also call a `geometric point’, perhaps slightly nonstandardly) is simply a map . 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 is affine, for example the elliptic curve over , , such a map corresponds to a map of rings

In other words, it corresponds to a choice of values for and in satisfying the governing equation – exactly what we should mean by specifying a -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 , 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 -rational point is associated with a unique closed point . In fact, we get an induced map , where is the residue field at . 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 determines the a set of -rational points which can be canonically identified with . But by basic Galois theory, this says that, if , is nonempty iff , in which case .

So let us return to our zeta function. Writing , we can expand

Now, set and bearing in mind where we are trying to get (to an expression involving the numbers ), we can define to be if and otherwise, and rewrite this sum as

Now, take the sum over all closed points , and conclude that

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 over all the finite field extensions of , 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 over a finite field asserts that all the zeroes are on the lines and the poles are on the lines . Noting that is an object of dimension 1, the analogy with the classical Riemann hypothesis could not be clearer.

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