Related papers: Constructing curves over finite fields with many p…
In this paper we give several methods to construct curves over finite fields with many points and illustrate this with examples of the results.
We construct curves with many points over finite fields using the class group
The problem of constructing curves with many points over finite fields has received considerable attention in the recent years. Using the class field theory approach, we construct new examples of curves ameliorating some of the known…
Elliptic curves over finite fields with predefined conditions in the order are practically constructed using the theory of complex multiplication. The stage with longest calculations in this method reconstructs some polynomial with integer…
I provide methods of constructing elliptic and hyperelliptic curves over global fields with interesting rational points over the given fields or over large field extensions. I also provide a elliptic curves defined over any given number…
Let $\mathbb F_{q^2}$ be the finite field with $q^2$ elements. We provide a simple and effective method, using reciprocal polynomials, for the construction of algebraic curves over $\mathbb F_{q^2}$ with many rational points. The curves…
We use class field theory to search for curves with many rational points over small finite fields. By going through abelian covers of curves of small genus we find a number of new curves. In particular, we settle the question of how many…
We construct pairs of elliptic curves over number fields with large intersection of projective torsion points.
We present a framework for constructing examples of smooth projective curves over number fields with explicitly given elements in their second K-group using elementary algebraic geometry. This leads to new examples for hyperelliptic curves…
In this paper we study a family of curves obtained by fibre products of hyperelliptic curves. We then exploit this family to construct examples of curves of given genus g over a finite field Fq with many rational points. The results…
We give a construction of singular curves with many rational points over finite fields. This construction enables us to prove some results on the maximum number of rational points on an absolutely irreducible projective algebraic curve…
We present a method for constructing optimized equations for the modular curve X_1(N) using a local search algorithm on a suitably defined graph of birationally equivalent plane curves. We then apply these equations over a finite field F_q…
The defect of a curve over a finite field is the difference between the number of rational points on the curve and the Weil-Serre upper bound for the number of points on the curve. We present algorithms for constructing curves of genus 5,…
We describe a method that allows, under some hypotheses, to compute all the rational points of some genus 5 curves defined over a number field. This method is used to solve some arithmetic problems that remained open.
We develop a constructive process which determines all extreme points of the unit ball of the space of $m$--linear forms, $m\geq1.$ Our method provides a full characterization of the geometry of that space through finitely many elementary…
We give a method for constructing Kummer covers with many points over finite fields.
We present an algorithm for detecting basepoints of linear series of curves in the plane. Moreover, we give an algorithm for constructing a linear series of curves in the plane for given basepoints. The underlying method of these algorithms…
We show how to construct Darmon points on elliptic curves over totally real fields.
This is a survey paper dealing with moduli aspects of curves over finite fields. It discusses counting points of moduli spaces, relations with modular forms and stratifications on moduli spaces.
In this paper we present an iterative construction of irreducible polynomials over finite fields based upon repeated applications of transforms induced by endomorphisms of odd prime degree of ordinary elliptic curves.