Conference "Arithmetic, Geometry and Coding Theory"

March 14 - 18, 2011

CIRM, Marseille, France

Organisers: Yves Aubry (Université de la Méditerranée, IML), Christophe Ritzenthaler (Université de la Méditerranée, IML), Alexey Zykin (Mathematical Department of HSE, Laboratoire Poncelet, IITP)

On the number of rational points on algebraic curves over finite fields

Henning Stichtenoth

Abstract

For a curve C over a finite field Fq (projective, nonsingular, absolutely irreducible) we denote by g(C) (resp. N(C)) the genus (resp. the number of F_q-rational points) of C. The classical Hasse-Weil Theorem says that for given q and g = g(C), q + 1 - 2g \sqrt{q} \leq N(C) \leq q + 1 + 2g \sqrt{q}; i.e. N(C) lies in a finite interval. A lot of effort has been put into improving the upper bound of this interval, partly motivated by applications of curves with `many' points in coding theory and cryptography, but also since `the question represents an attractive mathematical challenge' (van der Geer). In this talk, we change the point of view slightly: we fix the finite field F_q and a non-negative integer N and ask for all possible values of g such that there exists a curve C over F_q of genus g, having exactly N rational points. As follows immediately from the Hasse-Weil Theorem, g must satisfy the condition g \geq (N - q - 1)/ (2\sqrt{q}) Our main result is
Theorem 1. Given a finite field F_q and an integer N \geq 0, there is an integer g0 \geq 0 such that for every g \geq g0, there exists a curve C over F_q with g(C) = g and N(C) = N.
A generalization of this result is
Theorem 2. Given a finite field F)q and non-negative integers b_1,..., b_m. Then there exists an integer g_1 with the following property: for every g \geq g_1 there exists a curve C over F_q with g(C) = g such that C has exactly b_r points of degree r, for r = 1,..., m.
Theorem 2 has a nice interpretation in terms of the L-polynomial (the numerator of the zeta function) of the curve C. One can - under certain conditions - prescribe the first coefficients of the L-polynomial of C arbitrarily.

Web page of the conference at CIRM