Curves of fixed gonality with many rational points

The Newton Polygon of Plane Curves with Many Rational Points

This curve has the points (1 : 0 : 0) and (0 : 1 : 0) at infinity over any field. The affine equation is XY +Y +X = 0. The origin is a point of this curve. If (x, y) ∈ F8 is a point of this curve with nonzero coordinates, then x = 1. So 0 = xy + y + x = xy + xy + x = x[(xy) + (xy) + 1]. Let t = xy. Then t + t+ 1 = 0. So the Klein quartic has 3.7 = 21 rational points over F8 with nonzero coordin...

On Curves over Finite Fields with Many Rational Points

We study arithmetical and geometrical properties of maximal curves, that is, curves defined over the finite field Fq2 whose number of Fq2 -rational points reachs the Hasse-Weil upper bound. Under a hypothesis on non-gaps at rational points we prove that maximal curves are Fq2 -isomorphic to y q + y = x for some m ∈ Z.

Algebraic curves over F2 with many rational points

A smooth, projective, absolutely irreducible curve of genus 19 over F2 admitting an infinite S-class field tower is presented. Here S is a set of four F2-rational points on the curve. This is shown to imply that A(2) = limsup#X(F2)/g(X) ≥ 4/(19 − 1) ≈ 0.222. Here the limit is taken over curves X over F2 of genus g(X)→∞.

Curves with many points

Introduction. Let C be a (smooth, projective, absolutely irreducible) curve of genus g > 2 over a number field K. Faltings [Fa1, Fa2] proved that the set C(K) of K-rational points of C is finite, as conjectured by Mordell. The proof can even yield an effective upper bound on the size #C(K) of this set (though not, in general, a provably complete list of points); but this bound depends on the ar...

Rational points on curves