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…)