Related papers: Computing all S-integral points on elliptic curves
In this note we give an algorithm to explicitly construct the modular parametrization of an elliptic curve over the rationals given the Weierstrass function $\wp (z)$.
We give the complete list of possible torsion subgroups of elliptic curves with complex multiplication over number fields of degree 1-13. Additionally we describe the algorithm used to compute these torsion subgroups and its implementation.
We derive a priori estimates for solutions of a general class of fully non-linear equations on compact Hermitian manifolds. Our method is based on ideas that have been used for different specific equations, such as the complex…
In this short note, we give a method for computing a non-torsion point of smallest canonical height on a given elliptic curve $E/\mathbb{Q}$ over all number fields of a fixed degree. We then describe data collected using this method, and…
We contribute a new algebraic method for computing the orthogonal projections of a point onto a rational algebraic surface embedded in the three dimensional projective space. This problem is first turned into the computation of the finite…
We extend the explicit quadratic Chabauty methods developed in previous work by the first two authors to the case of non-hyperelliptic curves. This results in an algorithm to compute the rational points on a curve of genus $g \ge 2$ over…
This is an introduction to a probabilistic model for the arithmetic of elliptic curves, a model developed in a series of articles of the author with Bhargava, Kane, Lenstra, Park, Rains, Voight, and Wood. We discuss the theoretical evidence…
We describe an algorithm to compute the local Coleman-Gross p-adic height at p on a hyperelliptic curve. Previously, this was only possible using an algorithm due to Balakrishnan and Besser, which was limited to odd degree. While we follow…
We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational…
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…
We generalize a result of Frey [Fre88] on the Selmer group of twists of elliptic curves over Q with Q-rational torsion points to elliptic curves defined over number fields of small degree K with a K-rational point. We also provide examples…
The Langlands Programme predicts that a weight 2 newform f over a number field K with integer Hecke eigenvalues generally should have an associated elliptic curve E_f over K. In our previous paper, we associated, building on works of Darmon…
We show how the size of the Galois groups of iterates of a quadratic polynomial $f(x)$ can be parametrized by certain rational points on the curves $C_n:y^2=f^n(x)$ and their quadratic twists. To that end, we study the arithmetic of such…
A new algorithm for computing a point on a polynomial or rational curve in B\'{e}zier form is proposed. The method has a geometric interpretation and uses only convex combinations of control points. The new algorithm's computational…
We compute the $L$-functions of a large class of algebraic curves, and verify the expected functional equation numerically. Our computations are based on our previous results on stable reduction to calculate the local $L$-factor and the…
Baker's method, relying on estimates on linear forms in logarithms of algebraic numbers, allows one to prove in several situations the effective finiteness of integral points on varieties. In this article, we give a generalisation of…
In this paper, we describe a general method for computing Selberg-like integrals based on a formula, due to Kaneko, for Selberg-Jack integrals. The general principle consists in expanding the integrand \emph{w.r.t.} the Jack basis, which is…
We derive the complete and optimal Cheng--Yau gradient estimates and universal bounds for subcritical semilinear elliptic equations on Riemannian manifolds with (Bakry-\'{E}mery) Ricci curvature bounded below. This answers a fundamental…
We present new ideas for computing elliptic Gau{\ss} sums, which constitute an analogue of the classical cyclotomic Gau{\ss} sums and whose use has been proposed in the context of counting points on elliptic curves and primality tests. By…
We analyse and drastically improve the running time of the algorithm of Mazur, Stein and Tate for computing the canonical cyclotomic p-adic height of a point on an elliptic curve E/Q, where E has good ordinary reduction at p >= 5.