Related papers: Uniform bounds for rational points on complete int…
In this short note we show that the uniform abc-conjecture over number fields puts strong restrictions on the coordinates of rational points on elliptic curves. For the proof we use a variant of the uniform abc-conjecture over number fields…
We show that measures of irrationality on very general codimension two complete intersections and very general complete intersection surfaces are multiplicative in the degrees of the defining equations. This confirms some cases of a…
We apply the specialization technique based on the decomposition of the diagonal to find an explicit example over $\mathbb{Q}$ of a quadric and cubic hypersurface in $\mathbb{P}^6$ such that their intersection is a smooth stably irrational…
We study rationality properties of smooth complete intersections of three quadrics in $\mathbb{P}^7$. We exhibit a smooth family of such intersections with both rational and non-rational fibers.
We consider diagonal cubic surfaces defined by an equation of the form ax^3+by^3+cz^3+dt^3 = 0. Numerically, one can find all rational points of height < B for B in the range of up to 100 000, thanks to a program due to D. J. Bernstein. On…
We prove asymptotic formulas for the number of rational points of bounded height on certain equivariant compactifications of the affine plane.
We give upper and lower bounds for the number of rational points on Prym varieties over finite fields. Moreover, we determine the exact maximum and minimum number of rational points on Prym varieties of dimension 2.
We prove asymptotic formulas for the number of rational points of bounded height on certain blow-ups of the projective space.
In this paper we give an upper bound on the number of rational points on an irreducible curve $C$ of degree $\delta$ defined over a finite field $\mathbb{F}_q$ lying on a Frobenius classical surface $S$ embedded in $\mathbb{P}^3$. This…
In this paper we compute upper bounds for the number of ordinary triple points on a hypersurface in $P^3$ and give a complete classification for degree six (degree four or less is trivial, and five is elementary). But the real purpose is to…
Caro and Pasten gave an explicit upper bound on the number of rational points on a hyperbolic surface that is embedded in an abelian variety of rank at most one. We show how to use their method to produce a refined bound on the number of…
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 prove two related concentration inequalities concerning the number of rational points of hyperelliptic curves over subsets of a finite field. In particular, we investigate the probability of a large discrepancy between the numbers of…
Let $n$ be a positive multiple of $4$. We establish an asymptotic formula for the number of rational points of bounded height on singular cubic hypersurfaces $S_n$ defined by $$ x^3=(y_1^2 + \cdots + y_n^2)z . $$ This result is new in two…
An upper bound of the expected order of magnitude is established for the number of rational points of bounded height on Ch\^atelet surfaces defined over the rationals.
For non-singular intersections of pairs of quadrics in 11 or more variables, we prove an asymptotic for the number of rational points in an expanding box.
Let $X_4\subset\mathbb{P}^{n+1}$ be a quartic hypersurface of dimension $n\geq 4$ over an infinite field $k$. We show that if either $X_4$ contains a linear subspace $\Lambda$ of dimension $h\geq \max\{2,\dim(\Lambda\cap…
Among the nondegenerate C^4 hypersurfaces M in R^n, we characterize the rational quadrics as the hypersurfaces that are the least well approximated by rational points. Given M other than a rational quadric, we prove a heuristically sharp…
An asymptotic formula is established for the number of rational points of bounded height on a non-singular quartic del Pezzo surface with a conic bundle structure.
In 1991 S{\o}rensen proposed a conjecture for the maximum number of points on the intersection of a surface of degree $d$ and a non-degenerate Hermitian surface in $\PP^3(\Fqt)$. The conjecture was proven to be true by Edoukou in the case…