Abstract

How many points are there on a curve with coordinates in a given finite field when the curve has (a) no singular points or (b) singular points counted once or (c) singular points counted with multiplicity?

What is the maximum number of points on a curve of given genus?

Can curves attaining this maximum number be characterised?

Introduction

Problems in combinatorics, especially in finite geometry, often require a count of the number of solutions of an equation in one or more unknowns defined over a finite field Fq. When two unknowns, say X, Y, occur, the equation is of type f(X, Y) = 0 with f ∈ Fq[X, Y], and the geometric approach for solving it depends on the theory of algebraic curves over finite fields.

Curves over a finite field have applications in the theory of linear error-correcting codes in two areas: (a) the construction of Goppa or algebraic-geometry codes; (b) obtaining bounds for the maximum length of codes when given the dimension and minimum distance.

In cryptography, ciphers are constructed from both elliptic and hyperelliptic curves

It is natural to think about a plane algebraic curve F of equation f(X, Y) = 0 as the set of the points P = (x, y) in the affine plane over the coordinate field K such that f(x, y) = 0. But important numerical results on curves and their intersections, such as Bézout's theorem, have an easier formulation when the following are taken into consideration.