The Weil bound for curves over finite fields

The purpose of this post is to describe an application of the general intersection theory machinery (for curves on surfaces) developed in the previous posts: the Weil bound on points on a curve over a finite field.

1. Statement of the Weil bound

Let ${C}$ be a smooth, projective, geometrically irreducible curve over ${\mathbb{F}_q}$ of genus $g$ . Then the Weil bound states that:

$\displaystyle |C(\mathbb{F}_q) - q - 1 | \leq 2 g \sqrt{q}.$

Weil’s proof of this bound is based on intersection theory on the surface ${C \times C}$ . More precisely, let

$\displaystyle \overline{C} = C \times_{\mathbb{F}_q} \overline{\mathbb{F}_q},$

so that ${\overline{C}}$ is a smooth, connected, projective curve. It comes with a Frobenius map

$\displaystyle F: \overline{C} \rightarrow \overline{C}$

of ${\overline{\mathbb{F}_q}}$ -varieties: in projective coordinates the Frobenius runs

$\displaystyle [x_0: \dots : x_n] \mapsto [x_0^q: \dots : x_n^q].$

In particular, the map has degree ${q}$ . One has

$\displaystyle C( \mathbb{F}_q) = \mathrm{Fix}(F, \overline{C}(\overline{\mathbb{F}}_q))$

representing the ${\mathbb{F}_q}$ -valued points of ${C}$ as the fixed points of the Frobenius (Galois) action on the ${\overline{\mathbb{F}_q}}$ -valued points. So the strategy is to count fixed points, using intersection theory.

Using the (later) theory of ${l}$ -adic cohomology, one represents the number of fixed points of the Frobenius as the Lefschetz number of ${F}$ : the action of ${F}$ on ${H^0}$ and ${H^2}$ give the terms ${q+1}$ . The fact that (remaining) action of ${F}$ on the ${2g}$ -dimensional vector space ${H^1}$ can be bounded is one of the Weil conjectures, proved by Deligne for general varieties: here it states that ${F}$ has eigenvalues which are algebraic integers all of whose conjugates have absolute value ${\sqrt{q}}$ .

2. Some divisors on the square of $C$

Let’s consider the divisor ${\Gamma}$ given by the image of the closed immersion

$\displaystyle \overline{C} \rightarrow \overline{C} \times \overline{C}, \quad x \mapsto (x, Fx) .$

In particular, ${\Gamma}$ is a smooth curve. The set-theoretic intersection of ${\Gamma}$ with the diagonal divisor ${\Delta}$ clearly consists of the fixed points of ${F}$ . Moreover,

$\displaystyle C( \mathbb{F}_q) = \Gamma. \Delta,$

because ${\Gamma}$ and ${\Delta}$ intersect transversely at all points where they intersect. The tangent space to ${\Gamma}$ is “horizontal” (the Frobenius has zero differential) while that to ${\Delta}$ is “diagonal.” So the main goal is to obtain bounds on the intersection pairing ${\Gamma.\Delta}$ . The Hodge index theorem provides a powerful tool in this regard.

Namely, we know that ${\Gamma}$ has the following property:

The intersection ${\Gamma. h}$ with the “horizontal” divisor ${C \times \left\{\ast\right\}}$ is given by ${q}$ , the degree of the Frobenius.
The intersection ${\Gamma. v}$ with the “vertical” divisor ${\left\{\ast\right\} \times C}$ is ${1}$ . More generally, this is true for the graph of any morphism.

This raises a more general question: given a divisor ${D}$ on ${\overline{C} \times \overline{C}}$ with known “horizontal” and “vertical” intersection numbers, can we bound ${D. \Delta}$ ? This is something that we can answer using the Hodge index theorem proved in the previous post.

Namely, let’s work in the group ${\mathrm{Num}(X)\otimes_{\mathbb{Z}} \mathbb{Q}}$ of rational divisors modulo numerical equivalence. We have an orthogonal decomposition

$\displaystyle D = a h + bv + D', \quad \Delta = h + v + \Delta' .$

Here ${D', \Delta' }$ is perpendicular to ${\mathbb{Q} h \oplus \mathbb{Q} v}$ . Using the relations ${h^2 = v^2 =0 , h.v = 1}$ , ${a,b}$ can be computed from the knowledge of the “horizontal” and “vertical” intersection numbers of ${D}$ via

$\displaystyle a = D.v, \quad b = D.h.$

In particular, we get

$\displaystyle D. \Delta = ( ah + bv + D').( h + v + \Delta') = a + b + D'.\Delta'.$

But now the Hodge index theorem tells us that the intersection pairing is negative definite on ${(\mathbb{Q}h \oplus \mathbb{Q} v)^{\perp}}$ —indeed, ${h+v}$ is an ample divisor, so that by Cauchy-Schwarz:

$\displaystyle |D' \Delta'| \leq \sqrt{ D'^2 \Delta'^2}. \ \ \ \ \ (1)$

Let’s now simplify this a bit. We had ${ \Delta^2 = 2-2g; }$ the self-intersection of the diagonal in ${\overline{C} \times \overline{C}}$ is the Euler characteristic. One can see this by observing that ${\Delta^2}$ is the degree of the normal bundle to the diagonal embedding ${\overline{C} \rightarrow \overline{C} \times \overline{C}}$ ; this is the tangent (or anticanonical) bundle to ${\overline{C}}$ , which has degree ${2-2g}$ .

A simple computation shows therefore

$\displaystyle \Delta'^2 = -2g, \quad D'^2 = D^2 - 2ab.$

Proposition 1 Let ${D}$ be any divisor on ${\overline{C} \times\overline{C}}$ with ${a = D.v, b = D.h}$ . Then

$\displaystyle |D. \Delta - (a+b)| \leq \sqrt{ 2g(2ab - D^2)}. \ \ \ \ \ (2)$

3. The Weil bound

Let’s now apply (2) and prove the Weil bound. Let’s now apply this to the case when ${D}$ is the graph ${\Gamma}$ of the Frobenius. For ${\Gamma^2}$ , we get ${q(2 - 2g)}$ . The normal bundle to the imbedding

$\displaystyle \overline{C} \stackrel{x \mapsto (x, Fx)}{\rightarrow} \overline{C} \times \overline{C}$

is exactly ${F^* T_{\overline{C}}}$ , which has degree ${q(2-2g)}$ .

In this case,

$\displaystyle a = q, b = 1, \Gamma^2 = q(2 - 2g),$

and we conclude with:

Theorem 2 (Weil bound) For the smooth projective curve ${C/\mathbb{F}_q}$ , one has

$\displaystyle |C(\mathbb{F}_q) - q - 1| \leq 2 g \sqrt{q}. \ \ \ \ \ (3)$

Climbing Mount Bourbaki