Wan: Partial Counting of Rational Points over Finite Fields

We are motivated by the following problem. Let , where is irreducible of degree over . We look at the group

where . When does , for example?

Let , and let

By a character sum argument counting, you can write this as

By the Riemann hypothesis (Weil), we have . Therefore we have seen:

Proposition. Let . Let . Then

If , then

In particular, if , then and .

Therefore we consider the problem: Can we compute in time polynomial in , , and ?

The general setup: Let , and . We want to count

Can we compute , or at least estimate it? How does this quantity vary when the vary?

For example, we consider the Artin-Schreier hypersurface. Let

where . For each , we consider

Heuristically (for suitable ), we expect

where the constant depends on , , and .

Theorem. [Deligne] Write , where are homogeneous of degree . Assume defines a smooth projective hypersurface in , and that , . Then

Definition. If , we define the th fibred sum of to be

Theorem. [Fu-W] Write , and assume that is smooth in and . Then

Example. In the case that we can write

and is smooth in , is smooth in . Then is smooth in if and only if .

Since the condition that the fibred sum be smooth is Zariski open, we have shown it is nonempty if and therefore there exist many examples of such to which the theorem applies.

Definition. Let be the set of over such that is smooth. Then is Zariski open in the set of all over with .

Theorem. [Gao-W] is Zariski dense if and only if . In fact,

and this intersection is Zariski open and dense.

where and ?

Remark. We expect , but one can get the weaker estimate in many cases (Katz).

Now we consider partial zeta functions over . Let , . Define

where

Without loss of generality, we may assume , since otherwise we can just enlarge the ground field .

Proposition.

1. If , then .

2. (Faltings) , where and is a primitive th root of unity, , and .

By exponentiation to a root of unity, we mean the formal binomial expansion. From a counting point of view, this is as good as rational'.

Theorem. In all cases, .

Proof. Let . We have a map

Faltings constructs a large subvariety which is stable under . Then

Note . This implies Faltings' near' rationality as in the proposition.

Now . We refine the above argument as follows. First, for , you have

We consider , where . We have a character , and define the -function

By Grothendieck, . This implies that

so (essentially by unique factorization).

Open problem: can you bound the total degree of ? The best bound we have is , . Can this be improved to ? Yes, if and (Fu-W).

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