# Zeta functions

Here’s the rough idea: is a set of equations in a finite number of variables over the integers. Set of solutions mod p. The goal is to determine formulas that calculate .

Translating this to schemes, let be a scheme of finite type over . Then set = cardinality of maps from into X. For example, for for some , then counts the number of square roots for .

Here’s an explicit example. . To give a map from, for example, into X is equivalent to give a ring map into the field which requires a choice of where to send x, such that has a square root in the field. Now for is and these latter numbers have square roots respectively. Thus . As it happens .

### Definitions and Easy properties

Generalizing a little bit, let be a scheme of finite type over a finite field . Then define

I.e. the number of maps from into X in the category of – schemes.

Here are some properties (all unions are meant to be disjoint). If then . In what follows I write for . Its straightforward to see that so . Now using

I find , for .

Some notation. Let . Another important example: for fixed e. Now the elements of correspond to ring homomorphism

Its easy to see the map must be injective. If , then , so . Further, must now be a root of . Meaning which happens iff . In this case the number of such is equal to the order of the Galois group which is because its a Galois extension. Putting this all together I have

From the previous post this is

Where are the e-th roots of unity.

### The (local) zeta function

For set

Also introduce the numbers

Where are closed points and .

### A bunch of results

In the future I might add some of the proofs of these things.

- If is of finite type of dim = m, then there is a constant c s.t for all n.
- If dim X is positive then is unbounded.

Here’s a big theorem I wont prove

Thm: There exists a finite number of s.t.

In view of the last post, this says

Further, if the hypothesis of #3 are satisfied then converges for

—————————-

The proof of #3 uses Noether Normalization and CRT; #4 uses the Nulstullenzats (spelling?) and the last statement of the theorem follows from #3 and nth root formula for radius of convergence.

### More Results

Thm: Let be a smooth geometrically connected projective curve of genus g.

- ,
- is a permutation of the and
- for all i
- The residue of at is the order of a certain finite group having to do with (I might actually explain this later).

Cor. so that #3 also implies .

One final thing. There’s a degree homomorphism . In general the image is of the form for . For a curve as the thm above Ogus proved that as follows:

Lemma1: where f is a poly and so the Z(T) has simple poles at .

Lemma2: If and then .

Lemma2 is not so bad to prove, all the work is in understanding why .

In any case, continuing to show e = 1, using lemma 1

So lemma 2 gives

But by lemma 1 the zeta function has to have simple poles, so e = 1.

## About this entry

You’re currently reading “Zeta functions,” an entry on Math Meandering

- Published:
- August 18, 2009 / 3:49 pm

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- Ogus Excerpts

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