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 ${\sqrt{q}}$. (more…)