Related papers: Rational points on diagonal quartic surfaces

Let $f(x)=x^5+ax^3+bx^2+cx \in \Z[x]$ and consider the hypersurface of degree five given by the equation \cal{V}_{f}: f(p)+f(q)=f(r)+f(s). Under the assumption $b\neq 0$ we show that there exists $\Q$-unirational elliptic surface contained…

Let X be a smooth quartic surface not containing lines, defined over a number field K. We prove that there are only finitely many bitangents to X which are defined over K. This result can be interpreted as saying that a certain surface,…

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…

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…

We compute the number of rational quartics on a general Calabi-Yau hypersurface in weighted projective space P(2,1^4). The result agrees with the prediction made by mirror symmetry.

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 $X\subset \mathbb{P}^{n-1}_{K}$…

A quadrilateral is said to be rational if its four sides, the two diagonals and the area are all expressible by rational numbers. The problem of constructing rational quadrilaterals dates back to the seventh century when Brahmagupta gave an…

We survey some results on real rational surfaces focused on their topology and their birational geometry.

In this paper,we count the rational points on the weighted projective spaces defined over number fields w.r.t. ``size''. An asymptotic formula which generalizes the result of Schanuel's ``Heights in number fields'' is obtained. Furthermore,…

We show that, for every prime number p, there exist infinitely many K3 surfaces over Q whose rational points lie dense in the space of p-adic points. We also show that there exists a K3 surface over Q whose rational points lie dense in the…

We consider the question of how well points in a quadric hypersurface $M\subset\mathbb R^d$ can be approximated by rational points of $\mathbb Q^d\cap M$. This contrasts with the more common setup of approximating points in a manifold by…

Given a non-singular diagonal cubic hypersurface $X\subset\mathbb{P}^{n-1}$ over $\mathbb{F}_q(t)$ with $\mathrm{char} (\mathbb{F}_q)\neq 3$, we show that the number of rational points of height at most $|P|$ is $O(|P|^{3+\varepsilon})$ for…

A conjecture of Serre concerns the number of rational points of bounded height on a finite cover of projective space P^{n-1}. In this paper, we achieve Serre's conjecture in the special case of smooth cyclic covers of any degree when n is…

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 $\Fq$ be the finite field with $q$ elements. We study the number of $\Fq$-rational points on Danielewski and double Danielewski surfaces. For Danielewski surfaces, the point count is reduced to the number of roots of $P(Z)$ over $\Fq.$…

Motivated by a recent question of Peyre, we apply the Hardy-Littlewood circle method to count "sufficiently free" rational points of bounded height on arbitrary smooth projective hypersurfaces of low degree that are defined over the…

We investigate the Hasse principle for complete intersections cut out by a quadric and cubic hypersurface defined over the rational numbers.

We consider the practical computation of rational points on y^2=x(x^2+ax+b). The algebra necessary for a 4-descent procedure is described. A simple further descent is then described which only uses integer arithmetic. Numerous examples are…

In this paper, we prove a general result computing the number of rational points of bounded height on a projective variety $V$ which is covered by lines. The main technical result used to achieve this is an upper bound on the number of…

Building on recent work of Bhargava--Elkies--Schnidman and Kriz--Li, we produce infinitely many smooth cubic surfaces defined over the field of rational numbers that contain rational points.