Related papers: Rational points near planar flat curves
Under fairly natural assumptions, Huang counted the number of rational points lying close to an arc of a planar curve. He obtained upper and lower bounds of the correct order of magnitude, and conjectured an asymptotic formula. In this…
We find an asymptotic formula for the number of rational points near planar curves. More precisely, if $f:\mathbb{R}\rightarrow\mathbb{R}$ is a sufficiently smooth function defined on the interval $[\eta,\xi]$, then the number of rational…
Deng (arXiv:math/9812082) gave an asymptotic formula for the number of rational points on a weighted projective space over a number field with respect to a certain height function. We prove a generalization of Deng's result involving a…
We provide an asymptotic estimate for the number of rational points of bounded height on a non-singular conic over the rationals. The estimate is uniform in the coefficients of the underlying quadratic form.
We study a particular plane curve over a finite field whose normalization is of genus 0. The number of rational points of this curve achieves the Aubry-Perret bound for rational curves. The configuration of its rational points and a…
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.
In this paper, we establish asymptotic formulae with optimal errors for the number of rational points that are close to a planar curve, which unify and extend the results of Beresnevich-Dickinson-Velani and Vaughan-Velani. Furthermore, we…
Given a smooth cubic hypersurface $X$ over a finite field of characteristic greater than 3 and two generic points on $X$, we use a function field analogue of the Hardy-Littlewood circle method to obtain an asymptotic formula for the number…
We prove upper bounds for the number of rational points on non-singular cubic curves defined over the rationals. The bounds are uniform in the curve and involve the rank of the corresponding Jacobian. The method used in the proof is a…
Let $\mathbb{F}_q$ denote the finite field with $q$ elements. In this work, we use characters to give the number of rational points on suitable curves of low degree over $\mathbb{F}_q$ in terms of the number of rational points on elliptic…
In this article we establish an asymptotic formula for the number of rational points, with bounded denominators, within a given distance to a compact submanifold $\mathcal{M}$ of $\mathbb{R}^M$ with a certain curvature condition. Our result…
In this notes, we will give some remarks on the results in Rational points of universal curves by Hain. In particular, we consider the universal curves $\mathcal{M}_{g,n+1}\to \mathcal{M}_{g,n}$ and the sections of their algebraic…
In characteristic $p>0$ and for $q$ a power of $p$, we compute the number of nonplanar rational curves of arbitrary degrees on a smooth Hermitian surface of degree $q+1$ under the assumption that the curves have a parametrization given by…
Let $\mathscr{M}$ be a compact submanifold of $\mathbb{R}^{M}$. In this article we establish an asymptotic formula for the number of rational points within a given distance to $\mathscr{M}$ and with bounded denominators under the assumption…
We prove asymptotic formulas for the number of rational points of bounded height on certain equivariant compactifications of the affine plane.
The number A(q) shows the asymptotic behaviour of the quotient of the number of rational points over the genus of non-singular absolutely irreducible curves over a finite field Fq. Research on bounds for A(q) is closely connected with the…
We consider the structure of rational points on elliptic curves in Weierstrass form. Let x(P)=A_P/B_P^2 denote the $x$-coordinate of the rational point P then we consider when B_P can be a prime power. Using Faltings' Theorem we show that…
We establish the sharp estimate <<_d N^{2/d} for the number of rational points of height at most N on an irreducible projective curve of degree d. We deduce this from a result for general hypersurfaces that is sensitive to the coefficients…
The number of rational points of a plane non-singular algebraic curve X defined over a finite field is computed, provided that the generic point of X is not an inflexion and that X is Frobenius non-classical with respect to conics.
We improve a bound due to the second author on number of rational points on smooth surfaces in $\mathbb{P}^3$ over finite fields and look at families of surfaces that achieve or nearly achieve this bound, for which we compute their exact…