Related papers: Uniformly counting rational points on conics

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.

This paper is concerned with the density of rational points of bounded height lying on a variety defined by an integral quadratic form Q. In the case of four variables, we give an estimate that does not depend on the coefficients of Q. For…

We give uniform upper bounds for the number of rational points of height at most $B$ on non-singular complete intersections of two quadrics in $\mathbb{P}^3$ defined over $\mathbb{Q}$. To do this, we combine determinant methods with descent…

We give an upper bound for the number of rational points of height at most $B$, lying on a surface defined by a quadratic form $Q$. The bound shows an explicit dependence on $Q$. It is optimal with respect to $B$, and is also optimal for…

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…

In this paper, we give a uniform upper bound on the rational points of bounded height provided by conics in a cubic surface. For this target, we give a generalized version of the global determinant method of Salberger by Arakelov geometry.

We examine the counting function for rational points on conics, and show how the point where the asymptotic behaviour begins depends on the size of the smallest zero.

Upper and lower bounds, of the expected order of magnitude, are obtained for the number of rational points of bounded height on any quartic del Pezzo surface over $\mathbb{Q}$ that contains a conic defined over $\mathbb{Q}$.

An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariski open subset of an arbitrary smooth biquadratic hypersurface in sufficiently many variables. The proof uses…

We count rational points of bounded height on the Cayley ruled cubic surface and interpret the result in the context of general conjectures due to Batyrev and Tschinkel.

We use a global version of Heath-Brown's $p-$adic determinant method developed by Salberger to give upper bounds for the number of rational points of height at most $B$ on non-singular cubic curves defined over $\mathbb{Q}$. The bounds are…

In this note, we establish an asymptotic formula for the number of rational points of bounded height on the singular cubic surface $$ x_0(x_1^2 + x_2^2)=x_3^3 $$ with a power-saving error term, which verifies the Manin-Peyre conjectures for…

We prove asymptotic formulas for the number of rational points of bounded height on certain equivariant compactifications of the affine plane.

We prove asymptotic formulas for the number of rational points of bounded height on smooth equivariant compactifications of the affine space. (Nous \'etablissons un d\'eveloppement asymptotique du nombre de points rationnels de hauteur…

We establish asymptotic formulas for counting rational points near finite type curves on the plane, generalizing Huang's result.

We prove asymptotic formulas for the number of rational points of bounded height on certain blow-ups of the projective space.

We consider intersections of n diagonal forms of degrees k 1 < $\bullet$ $\bullet$ $\bullet$ < kn, and we prove an asymptotic formula for the number of rational points of bounded height on these varieties. The proof uses the…

Let $F(x_1,...,x_n)$ be a form of degree $d\geq 2$, which produces a geometrically irreducible hypersurface in $\mathbb{P}^{n-1}$. This paper is concerned with the number of rational points on F=0 which have height at most $B$. Whenever…

We prove an asymptotic formula for the average number of rational points on Fano hypersurfaces that are contained in a small ball centered at a given adelic point. We also prove an asymptotic formula for the number of hypersurfaces…

In this paper we prove a formula for the number of rational points of bounded height relative to all the generators of the cone of effective divisor for a toric variety over a number field.