Related papers: Counting rational points on elliptic and hyperelli…

We show that transcendental curves in $\mathbb R^n$ (not necessarily compact) have few rational points of bounded height provided that the curves are well behaved with respect to algebraic sets in a certain sense and can be parametrized by…

Harron and Snowden counted the number of elliptic curves over $\mathbb{Q}$ up to height $X$ with torsion group $G$ for each possible torsion group $G$ over $\mathbb{Q}$. In this paper we generalize their result to all number fields and all…

We give the first examples of derived equivalences between varieties defined over non-closed fields where one has a rational point and the other does not. We begin with torsors over Jacobians of curves over Q and F_q(t), and conclude with a…

We develop geometry-of-numbers methods to count orbits in coregular vector spaces having bounded invariants over any global field. We apply these techniques to bound the average ranks and determine average Selmer group sizes of elliptic…

We prove that the number of rational points of bounded height on certain del Pezzo surfaces of degree 1 defined over Q grows linearly, as predicted by Manin's conjecture. Along the way, we investigate the average number of integral points…

We provide new explicit formulas for bounding the number of rational points on singular curves over finite fields. This enables us to obtain exact values of N q (g, $\pi$) which is defined as the maximum number of rational points over F q…

We introduce several new methods to obtain upper bounds on the number of solutions of the congruences $f(x) \equiv y \pmod p$ and $f(x) \equiv y^2 \pmod p,$ with a prime $p$ and a polynomial $f$, where $(x,y)$ belongs to an arbitrary square…

In the present article, we consider Algebraic Geometry codes on some rational surfaces. The estimate of the minimum distance is translated into a point counting problem on plane curves. This problem is solved by applying the upper bound…

Let $E$ be an elliptic curve over the rationals. We will consider the infinite extension $\mathbb{Q}(E_{\text{tor}})$ of the rationals where we adjoin all coordinates of torsion points of $E$. In this paper we will prove an explicit lower…

We give a structure theorem for the $m$-torsion of the Jacobian of a general superelliptic curve $y^m=F(x)$. We study existence of torsion on curves of the form $y^q=x^p-x+a$ over finite fields of characteristic $p$. We apply those results…

We prove that any smooth projective geometrically connected non-isotrivial curve of genus $g\ge 2$ over a one-dimensional function field of any characteristic has at most $16g^2+32g+124$ torsion points for any Abel-Jacobi embedding of the…

Let $K$ be an algebraically closed field of characteristic different from $2$, $g$ a positive integer, $f(x)\in K[x]$ a degree $2g+1$ monic polynomial without repeated roots, $C_f: y^2=f(x)$ the corresponding genus g hyperelliptic curve…

It was shown by Faltings and Hriljac that the N\'eron-Tate height of a point on the Jacobian of a curve can be expressed as the self-intersection of a corresponding divisor on a regular model of the curve. We make this explicit and use it…

We explain how recent work on 3-descent and 4-descent for elliptic curves over Q can be combined to search for generators of the Mordell-Weil group of large height. As an application we show that every elliptic curve of prime conductor in…

Faltings' theorem states that curves of genus $g \geq 2$ have finitely many rational points. Using the ideas of Faltings, Mumford, Parshin and Raynaud, one obtains an upper bound on the number of rational points, but this bound is too large…

We determine upper bounds on the number of rational points of an affine or projective algebraic set defined over an extension of a finite field by a system of polynomial equations, including the case where the algebraic set is not defined…

Results in $p$-adic transcendence theory are applied to two problems in the Chabauty-Coleman method. The first is a question of McCallum and Poonen regarding repeated roots of Coleman integrals. The second is to give lower bounds on the…

We generalize the lemmas of Thomas Kretschmer to arbitrary number fields, and apply them with a 2-descent argument to obtain bounds for families of elliptic curves over certain imaginary quadratic number fields with class number 1. One such…

We show that there are infinitely many elliptic curves $E/\mathbb{Q}$, up to isomorphism over $\overline{\mathbb{Q}}$, for which the finitely generated group $E(\mathbb{Q})$ has rank exactly $2$. Our elliptic curves are given by explicit…

Let G be the separable Galois group of a finite field F of characteristic p, and X/F an imaginary hyperelliptic curve such that G acts transitively on its set W(X) of Weierstrass points. The existence of a G-invariant 2-torsion point on the…