Related papers: Hunting for curves with many points
We use the relations between quadrics, trace codes and algebraic curves to construct algebraic curves over finite fields with many points and to compute generalized Hamming weights of codes.
A new family of maximal curves over a finite field is presented and some of their properties are investigated.
We classify completely the intersections of the Hermitian curve with parabolas in the affine plane. To obtain our results we employ well-known algebraic methods for finite fields and geometric properties of the curve automorphisms. In…
We define orbifold mapping class groups (with marked points) and study them using their action on certain orbifold analogs of arcs and simple closed curves. Moreover, we establish a Birman exact sequence for suitable subgroups of orbifold…
We address the problem of the maximal finite number of real points of a real algebraic curve (of a given degree and, sometimes, genus) in the projective plane. We improve the known upper and lower bounds and construct close to optimal…
We give explicit formulas for the number of points on reductions of elliptic curves with complex multiplication by any imaginary quadratic field. We also find models for CM $\mathbf{Q}$-curves in certain cases. This generalizes earlier…
We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques ([BP], [HBR]) and methods based on quasiorthogonality in the Mordell-Weil lattice…
Let $\mathcal{F}$ be a plane singular curve defined over a finite field $\mathbb{F}_q$. The linear system of plane curves of a given degree passing through the singularities of $\cF$ provides potentially good bounds for the number of points…
We present a very efficient algorithm to construct an elliptic curve E and a finite field F such that the order of the point group E(F) is a given prime number N. Heuristically, this algorithm only takes polynomial time Otilde((\log N)^3),…
We obtain a coarse relationship between geometric intersection numbers of curves and the sum of their subsurface projection distances with explicit quasi-constants. By using this relationship, we give applications in the studies of the…
One of the big questions in the area of curves over finite fields concerns the distribution of the numbers of points: Which numbers occur as the number of points on a curve of genus $g$? The same question can be asked of various subclasses…
The geometry of algebraic curves over finite fields is a rich area of research. In previous work, the authors investigated a particular aspect of the geometry over finite fields of the classical unit circle, namely how the number of…
We look for elliptic curves featuring rational points whose coordinates form two arithmetic progressions, one for each coordinate. A constructive method for creating such curves is shown, for lengths up to 5.
We construct finitely generated groups with strong fixed point properties. Let $\mathcal{X}_{ac}$ be the class of Hausdorff spaces of finite covering dimension which are mod-$p$ acyclic for at least one prime $p$. We produce the first…
We suggest a few projects for studying vertex algebras with emphasis on finite group viewpoints.
We develop a new form of patching that is both far-reaching and more elementary than the previous versions that have been used in inverse Galois theory for function fields of curves. A key point of our approach is to work with fields and…
In this paper, we present several methods for construction of elliptic curves with large torsion group and positive rank over number fields of small degree. We also discuss potential applications of such curves in the elliptic curve…
We reduce the classification of finite extensions of function fields (of curves over finite fields) with the same class number to a finite computation; complete this computation in all cases except when both curves have base field…
We study the problem of efficiently constructing a curve C of genus 2 over a finite field F for which either the curve C itself or its Jacobian has a prescribed number N of F-rational points. In the case of the Jacobian, we show that any…
We develop class field theory of curves over $p$-adic fields which extends the unramified theory of S. Saito. The class groups which approximate abelian \'etale fundamental groups of such curves are introduced in the terms of algebraic…