Related papers: Rational points on curves
A sieve for rational points on suitable varieties is developed, together with applications to counting rational points in thin sets, the number of varieties in a family which are everywhere locally soluble, and to the notion of friable…
We present an algorithm for computing curves and families of curves of prescribed degree and geometric genus on real rational surfaces.
We exhibit planar, rational curves of large degree over ${\mathbb F}_2$ that have a unique singular point, which has multiplicity 2. In characteristic 0 such curves exist only for degrees up to $6$. v.2: references updated and examples of…
The distribution of degree $d$ points on curves is well understood, especially for low degrees. We refine this study to include information on the Galois group in the simplest interesting case: $d = 3$. For curves of genus at least 5, we…
In this article, we study how to compute the number of $K$-rational points with a given $j$-invariant on an arbitrary modular curve. As an application, for each positive integer $n$, we determine the list of possible numbers of cyclic…
In this paper, we will give a uniform upper bound of the number of rational points of bounded height in non-singular curves by applying the global determinant method.
Let X be a projective cubic hypersurface of dimension 11 or more, which is defined over the rationals. In this paper it is shown that X contains rational points provided that the cubic form defining X can be written as the sum of two forms…
Several problems which could be thought of as belonging to recreational mathematics are described. They are all such that solutions to the problem depend on finding rational points on elliptic curves. Many of the problems considered lead to…
We build a database of genus 2 curves defined over $\mathbb Q$ which contains all curves with minimal absolute height $h \leq 5$, all curves with moduli height $\mathfrak h \leq 20$, and all curves with extra automorphisms in standard form…
This paper is concerned with configurations of points in a plane lattice which determine angles that are rational multiples of $\pi$. We shall study how many such angles may appear in a given lattice and in which positions, allowing the…
In this paper I demonstrate that any pair (m, n) of non-zero and distinct rational numbers may have, at most, four representations as the product of two rational factors such that the sum of factors of m coincides with the sum of factors of…
We show that the proportion of plane cubic curves over ${\mathbb Q}_p$ that have a ${\mathbb Q}_p$-rational point is a rational function in $p$, where the rational function is independent of $p$, and we determine this rational function…
In this article we use techniques from coding theory to derive upper bounds for the number of rational places of the function field of an algebraic curve defined over a finite field. The used techniques yield upper bounds if the…
In a recent paper, the author and St\"ohr established a bound on the number of iterated Frobenius pullbacks needed to transform a non-smooth non-decomposed point on a regular geometrically integral curve into a rational point. In this note…
We classify the graphs that can occur as the graph of rational preperiodic points of a quadratic polynomial over $\bold Q$, assuming the conjecture that it is impossible to have rational points of period $4$ or higher. In particular, we…
A Kloosterman refinement for function fields $K=\mathbb{F}_q(t)$ is developed and used to establish the quantitative arithmetic of the set of rational points on a smooth complete intersection of two quadrics $X\subset \mathbb{P}^{n-1}_{K}$…
The number $N_9(5)$, the maximal number of $\mathbb{F}_9$-rational points on curves over $\mathbb{F}_9$ of genus $5$ is unknown, but it is known that $32 \le N_9(5)\le 35$. In this paper, we enumerate hyperelliptic curves and trigonal…
The Oesterl\'e bound shows that a curve of genus 8 over the finite field $\mathbb{F}_4$ can have at most 24 rational points, and Niederreiter and Xing used class field theory to show that there exists such a curve with 21 points. We improve…
This article discusses two versions of elliptic equations obtained from a system of equations describing a rational cuboid. Analysis of elliptic equations shows that they are equivalent, and that there are rational points on the elliptic…
We present experimental evidence to support the widely held belief that one half of all elliptic curves have infinitely many rational points. The method used to gather this evidence is a refinement of an algorithm due to the author which is…