Related papers: Counting primitive points of bounded height

We study the arithmetic (geometric) progressions in the $x$-coordinates of quadratic points on smooth projective planar curves defined over a number field $k$. Unless the curve is hyperelliptic, we prove that these progressions must be…

We prove upper bounds on the number of rational points on transcendental curves in arbitrary $1$-h-minimal fields, similar to the Pila--Wilkie counting theorem in the o-minimal setting. These results extend results due to…

We find upper and lower bounds on the number of rational points that are $\psi$-approximations of some $n$-dimensional $p$-adic integer. Lattice point counting techniques are used to find the upper bound result, and a Pigeon-hole principle…

We count certain abelian surfaces with potential quaternionic multiplication defined over a number field $K$ by counting points of bounded height on some genus zero Shimura curves.

The present note is devoted to an amendment to a recent paper of Ellenberg, Lawrence and Venkatesh. Roughly speaking, the main result here states the subpolynomial growth of the number of integral points with bounded height of a variety…

In an earlier work, the first author and Petsche used potential theoretic techniques to establish a lower bound for the height of algebraic numbers that satisfy splitting conditions, such as being totally real or p-adic, improving on…

In this note, we continue to be interested in the relationship that connects the restricted distribution of finitude at the local level of intermediate fields of a purely inseparable extension $K/k$ to the absolute or global finitude of…

In this paper, we revisit some known results about stationary varifolds using simpler arguments. In particular, we obtain the height bound and the Lipschitz approximation along with its estimates, and as a consequence, the excess decay

We prove uniform upper bounds on the number of integral points of bounded height on affine varieties. If $X$ is an irreducible affine variety of degree $d\geq 4$ in $\mathbb{A}^n$ which is not the preimage of a curve under a linear map…

Let $X$ be a smooth projective curve over a number field $k$. Let $f\colon X \to \mathbb{P}^1$ be a non-constant morphism that realizes the gonality of $X$. In this article we study the growth rate of $\left\{P\in X\left(\overline{k}…

Let $L/K$ be a Galois extension of number fields. We prove two lower bounds on the maximum of the degrees of the irreducible complex representations of ${\rm Gal}(L/K)$, the sharper of which is conditional on the Artin Conjecture and the…

In this paper, we prove the first incidence bound for points and conics over prime fields. As applications, we prove new results on expansion of bivariate polynomial images and on certain variations of distinct distances problems. These…

We implement methods from the geometry of numbers to give explicit estimates for the number of integral ideals in a number field. We pay particular attention to minimising the effect of the degree $n$ of the number field on the error term…

Let $C\subset{\mathbb P}_K^2$ be an algebraic curve over a number field $K$, and denote by $d_K$ the degree of $K$ over ${\mathbb Q}$. We prove that the number of $K$-rational points of height at most $H$ in $C$ is bounded by $c…

Here we describe the distribution of rational points on the Hilbert scheme of two points in the projective plane. More specifically, we explicitly describe a two-parameter family of height functions $H_{s, t}$, such that the height function…

Given a field $k$ of characteristic zero and an indeterminate $T$, the main topic of the paper is the construction of specializations of any given finite extension of $k(T)$ of degree $n$ that are degree $n$ field extensions of $k$ with…

Considering the Teichm\"uller space of a surface equipped with Thurston's Lipschitz metric, we study geodesic segments whose endpoints have bounded combinatorics. We show that these geodesics are cobounded, and that the closest-point…

We give an upper bound on the number of rational points of an arbitrary Zariski closed subset of a projective space over a finite field. This bound depends only on the dimensions and degrees of the irreducible components and holds for very…

For points $(a,b)$ on an algebraic curve over a field $K$ with height $\mathfrak{h}$, the asymptotic relation between $\mathfrak{h}(a)$ and $\mathfrak{h}(b)$ has been extensively studied in diophantine geometry. When $K=\overline{k(t)}$ is…

Let us consider an algebraic function field defined over a finite Galois extension $K$ of a perfect field $k$. We give some conditions allowing the descent of the definition field of the algebraic function field from $K$ to $k$. We apply…