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 be a smooth, projective, geometrically irreducible curve over of genus . Then the **Weil bound** states that:

Weil’s proof of this bound is based on intersection theory on the surface . More precisely, let

so that is a smooth, connected, projective curve. It comes with a Frobenius map

of -varieties: in projective coordinates the Frobenius runs

In particular, the map has degree . One has

representing the -valued points of as the fixed points of the Frobenius (Galois) action on the -valued points. So the strategy is to count fixed points, using intersection theory.

Using the (later) theory of -adic cohomology, one represents the number of fixed points of the Frobenius as the Lefschetz number of : the action of on and give the terms . The fact that (remaining) action of on the -dimensional vector space can be bounded is one of the Weil conjectures, proved by Deligne for general varieties: here it states that has eigenvalues which are algebraic integers all of whose conjugates have absolute value . (more…)