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)

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