Rational points and Galois points for a plane curve over a finite field

Around Sziklai's conjecture on the number of points of a plane curve over a finite field

Article history: Received 14 July 2008 Revised 9 February 2009 Available online 10 March 2009 Communicated by Neal Koblitz

Sziklai's conjecture on the number of points of a plane curve over a finite field III

In the paper [11], Sziklai posed a conjecture on the number of points of a plane curve over a finite field. Let C be a plane curve of degree d over Fq without an Fq-linear component. Then he conjectured that the number of Fq-points Nq(C) of C would be at most (d− 1)q+1. But he had overlooked the known example of a curve of degree 4 over F4 with 14 points ([10], [1]). So we must modify this conj...

Rational Points on Curves over Finite Fields

Preface These notes treat the problem of counting the number of rational points on a curve defined over a finite field. The notes are an extended version of an earlier set of notes Aritmetisk Algebraisk Geometri – Kurver by Johan P. Hansen [Han] on the same subject. In Chapter 1 we summarize the basic notions of algebraic geometry, especially rational points and the Riemann-Roch theorem. For th...

Distribution of Rational Points: A Survey

The fluctuations in the number of points on a hyperelliptic curve over a finite field

Article history: Received 4 May 2008 Revised 26 August 2008 Available online 21 October 2008 Communicated by J. Brian Conrey The number of points on a hyperelliptic curve over a field of q elements may be expressed as q + 1 + S where S is a certain character sum. We study fluctuations of S as the curve varies over a large family of hyperelliptic curves of genus g. For fixed genus and growing q,...