Rational points on algebraic curves in infinite towers of number fields

Acquisition of Rational Points on Algebraic Curves

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...

Geometric Methods for Improving the Upper Bounds on the Number of Rational Points on Algebraic Curves over Finite Fields

Currently, the best upper bounds on the number of rational points on an absolutely irreducible, smooth, projective algebraic curve of genus g defined over a finite field Fq come either from Serre’s refinement of the Weil bound if the genus is small compared to q, or from Oesterlé’s optimization of the explicit formulae method if the genus is large. This paper presents three methods for improvin...

On the Distribution of the Number of Points on Algebraic Curves in Extensions of Finite Fields

Let C be a smooth absolutely irreducible curve of genus g ≥ 1 defined over Fq, the finite field of q elements. Let #C(Fqn) be the number of Fqn -rational points on C. Under a certain multiplicative independence condition on the roots of the zetafunction of C, we derive an asymptotic formula for the number of n = 1, . . . , N such that (#C(Fqn) − q − 1)/2gq belongs to a given interval I ⊆ [−1, 1...

On the number of rational points on curves over finite fields with many automorphisms

Using Weil descent, we give bounds for the number of rational points on two families of curves over finite fields with a large abelian group of automorphisms: Artin-Schreier curves of the form y−y = f(x) with f ∈ Fqr [x], on which the additive group Fq acts, and Kummer curves of the form y q−1 e = f(x), which have an action of the multiplicative group Fq . In both cases we can remove a √ q fact...