Related papers: Solving S-unit, Mordell, Thue, Thue-Mahler and gen…

The purpose of this paper is to give some new Diophantine applications of modularity results. We use the Shimura-Taniyama conjecture to prove effective finiteness results for integral points on moduli schemes of elliptic curves. For several…

We discuss Mahler's work on Diophantine approximation and its applications to Diophantine equations, in particular Thue-Mahler equations, S-unit equations and S-integral points on elliptic curves, and go into later developments concerning…

In this paper, we explore several threads arising from our recent joint work on arithmetic holonomy bounds, which were originally devised to prove new irrationality results based on the method of Ap\'ery limits. We propose a new method to…

Most, if not all, unconditional results towards the abc-conjecture rely ultimately on classical Baker's method. In this article, we turn our attention to its elliptic analogue. Using the elliptic Baker's method, we have recently obtained a…

Let $T$ be a positive integer, and $\mathcal{M}$ be a finitely presented module over the Laurent polynomial ring $\mathbb{Z}_{/T}[X_1^{\pm}, \ldots, X_N^{\pm}]$. We consider S-unit equations over $\mathcal{M}$: these are equations of the…

Let $h(x,y)$ be a non-degenerate binary cubic form with integral coefficients, and let $S$ be an arbitrary finite set of prime numbers. By a classical theorem of Mahler, there are only finitely many pairs of relatively prime integers $x,y$…

We first recall the connection, going back to A. Thue, between rational approximation to algebraic numbers and integer solutions of some Diophantine equations. Next we recall the equivalence between several finiteness results on various…

We establish new upper bounds for the height of the S-integral points of an elliptic curve. This bound is explicitly given in terms of the set S of places of the number field K involved, but also in terms of the degree of K, as well as the…

We solve the Diophantine equation $Y^2=X^3+k$ for all nonzero integers $k$ with $|k| \leq 10^7$. Our approach uses a classical connection between these equations and cubic Thue equations. The latter can be treated algorithmically via lower…

Presentaremos una nueva demostraci\'on del teorema de Shafarevich sobre finitud de curvas el\'ipticas con buena reducci\'on fuera de un conjunto finito de primos dado. Esto da un nuevo punto de entrada a teoremas fundamentales de finitud…

Let $E$ be an elliptic curve over a number field $K$. Descent calculations on $E$ can be used to find upper bounds for the rank of the Mordell-Weil group, and to compute covering curves that assist in the search for generators of this…

Let $p, q$ be twin prime numbers with $q-p=2$ . Consider the elliptic curves E=E_\sigma: y^2 = x (x+\sigma p)(x+\sigma q) . (\sigma =\pm 1). E=E_\sigma is also denoted as E_+ or E_- when \sigma = +1or $-1.Here the Mordell-Weil group and the…

Using the Thue-Siegel method, we obtain effective improvements on Liouville's irrationality measure for certain one-parameter families of algebraic numbers, defined by equations of the type $(t-a)Q(t)+P(t)=0$. We apply these to some…

Let $K$ be a number field, let $S$ be a finite set of places of $K$ containing the archimedean places and let $\mu$, $\alpha_1,\alpha_2,\alpha_3$ be non--zero elements in $K$. Denote by $\OS$ the ring of $S$--integers in $K$ and by…

We describe an algorithm that computes explicit models of hyperelliptic Shimura curves attached to an indefnite quaternion algebra over Q and Atkin-Lehner quotients of them. It exploits Cerednik-Drinfeld's non-archimedean uniformisation of…

We establish new uniform height inequalities for rational points on higher-dimensional varieties, extending the classical Roth-Schmidt-Subspace paradigm to the Arakelov-theoretic setting. Our main result provides sharp bounds for heights…

In the late 1990's, Bremner conjectured that long arithmetic progressions among the $x$-coordinates of rational points of an elliptic curve $E$ over $\mathbb{Q}$ should force the rank of $E$ to be large. This conjecture (and a broad…

Let $\alpha$ be an algebraic number of degree $d\ge 3$ and let $K$ be the algebraic number field $\Q(\alpha)$. When $\varepsilon$ is a unit of $K$ such that $\Q(\alpha\varepsilon)=K$, we consider the irreducible polynomial $f_\varepsilon(X)…

We explore a number of problems related to the quadratic Chabauty method for determining integral points on hyperbolic curves. We remove the assumption of semistability in the description of the quadratic Chabauty sets…

We develop a rigorous theoretical framework for principal manifold estimation that recovers a latent low-dimensional manifold from a point cloud observed in a high-dimensional ambient space. Our framework accommodates manifolds with…