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

Let , and let

**Proposition. ***Let
. Let
. Then
*

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

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

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

Heuristically (for suitable ), we expect

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

What about ?

**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

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,
*

**Problem. **
What about Kummer hypersurfaces

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

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

**Proposition. **

- If
, then
.
- (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,
.
*

Faltings constructs a large subvariety which is stable under . Then

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

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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