Chapter 8 The Lang–Weil estimate
Original lecture date: October 23, 2019.
In this lecture, we discuss the Lang–Weil theorem, which uses the Riemann hypothesis for curves to give a partial result towards the Weil conjectures for higher-dimensional varieties.
Readings 8.0.1.
We follow the original presentation of Lang–Weil [83].
Section 8.1
Our first statement of the Lang–Weil theorem is the following.
Theorem 8.1.1. Lang–Weil 1.
Let \(X\) be a scheme of finite type over \(\FF_q\) of dimension \(n\text{.}\) Let \(c\) be the number of irreducible components of \(X\) of dimension \(n\) which are also geometrically irreducible. Then we have the estimate
where the constant for the big-\(O\) notation depends only on the geometry of \(X_{\overline{\FF_q}}\text{.}\)
Proof.
We induction on the dimension of \(n\text{,}\) the case \(n=0\) being straightforward (and the source of the constant \(c\)). For \(n>0\text{,}\) choose a projection map \(X \rightarrow S\) from \(X\) onto a scheme \(S\) of dimension \(n-1\text{,}\) such that \(X\) and \(S\) have the same value of \(c\text{.}\) We may then compute \(\#X(\FF_q)\) by summing over the fibers of the map over \(\FF_q\)-points. By the induction hypothesis, the number of summands is \(cq^{n-1} + Q(q^{n-3/2})\text{;}\) by the Weil conjectures for curves, the number of points on each fiber is \(q + O(q^{1/2})\) where the implied constant in the big-\(O\) notation can be bounded in terms of the geometry of the projection map. This yields the desired estimate.
Here we illustrate by an example that \(c\) should count only geometrically irreducible components.
Example 8.1.2.
Let \(X:=V(x^2+y^2)\subset \AAA^2_{\FF_q}\text{,}\) for a prime power \(q\) such that \(q\equiv 3\) (mod 4). By the quadratic residue criterion for \(-1\text{,}\)
Note that \(X_{\FF_{q^k}}\) has only one irreducible component for odd \(k\text{,}\) but this component is not geometrically irreducible because it splits after a base change. Hence \(c=0\) for this case which coincides with the above calculation. If \(k\) is even, then \(X_{\FF_{q^k}}\) is a disjoint union of two geometrically irreducible points, hence \(c=2\) which also agrees with the calculation.
Here is a more precise version of the theorem, whose proof we omit.
Theorem 8.1.3. Lang–Weil 2.
Let \(X\) be a geometrically irreducible projective variety of dimension n admitting an embedding \(X \hookrightarrow \PP^m\) of degree d. Then we have the following inequality
where \(A(n,m,d)\) is a constant dependent only on \(n,m,d\text{.}\)
Remark 8.1.4.
The shape of the estimate in Lang–Weil is in some sense best possible: the exponent of \(q\) in the error term cannot be reduced. For example, if \(X = C \times_{\FF_q} \AAA^{n-1}\text{,}\) then the error term for \(\#X(\FF_q)\) comes from the error term for \(\#C(\FF_q)\text{,}\) and this can certainly be as large as \(O(q^{1/2})\) (e.g., for elliptic curves).
On the other hand, under additional hypotheses one can hope for a better error estimate. For example, Let \(X\) be a smooth hypersurface in \(\PP^n\) over \(\FF_q\text{.}\) Then using the Weil conjectures plus some additional knowledge (the hard Lefschetz theorem in Weil cohomology; see Theorem 9.3.1 for the étale version), one can show that
where the first term is coming from the geometry of the ambient space and the second term is from the interesting middle cohomology \(H^{n-1}(X)\text{.}\)
As an intermediate case, consider what happens if we drop the smoothness hypothesis. Then the exponent in the error term is \(O(q^{(n+d-2)/2})\) where \(d\) is the dimension of the singular locus \(X^{\sing}\) (or \(-1\) if \(X^{\sing} = \emptyset\)).
All of these results can be made with completely explicit error terms. See [50].