Serre, Jean-Pierre Nombres de points des courbes algébriques sur \(F_ q\). (French) Zbl 0538.14016 Sémin. Théor. Nombres, Univ. Bordeaux I 1982-1983, Exp. No. 22, 8 p. (1983). The author gives an overview concerning upper bounds for the numbers of rational points on a curve of genus g, over a finite field of q elements. In general one has the bound given by Weil’s theorem. There are more precise results of the following nature: (i) asymptotic results: Fix q, and let \(g\to \infty\); (ii) results for \(q=2\); (iii) results for \(g=1,2,3\). Reviewer: G.Faltings Cited in 6 ReviewsCited in 23 Documents MSC: 14G15 Finite ground fields in algebraic geometry 14G05 Rational points 14H45 Special algebraic curves and curves of low genus 14N10 Enumerative problems (combinatorial problems) in algebraic geometry Keywords:Weil-estimates; finite ground field; numbers of rational points on a curve × Cite Format Result Cite Review PDF Full Text: EuDML Online Encyclopedia of Integer Sequences: Maximal number of rational points that a (smooth, geometrically irreducible) curve of genus 3 over the finite field GF(q) can have, where q is the n-th prime power >= 2.