Related papers: Finding Elliptic Curves With Many Integral Points

Let $C$ be an elliptic curve defined over $\mathbb Q$ by the equation $y^2=x^3+Ax+B$ where $A,B\in\mathbb Q$. A sequence of rational points $(x_i,y_i)\in C(\mathbb Q),\,i=1,2,\ldots,$ is said to form a sequence of consecutive squares on $C$…

We show that, conditional on Zywina's effective version of the Serre uniformity conjecture, there is a natural way to parameterize non-CM $\mathbb{Q}$-rational points on all modular curves in terms of the rational points on finitely many…

For a given elliptic curve, its associated $L$-function evaluated at $1$ is closely related to its real period. In this article, we generalize this principle to a rational curve. We count the rational points over all finite fields and use…

We describe deterministic and probabilistic algorithms to determine whether or not a given monic irreducible polynomial H in Z[X] is a Hilbert class polynomial, and if so, which one. These algorithms can be used to determine whether a given…

In this paper we consider smooth affine elliptic plane curves having one place at infinity. We identify them with elliptic projective plane curves having only one cusp as their singular points and meeting with the line at infinity only at…

For real irreducible algebraic curves of the seventh degree, there are 22 types of singular points of multiplicity six, 174 types of singular points of multiplicity five, and at least 182 types of singular points of multiplicity four. For…

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…

Assuming the Generalized Riemann Hypothesis, we design a deterministic algorithm that, given a prime p and positive integer m=o(sqrt(p)/(log p)^4), outputs an elliptic curve E over the finite field F_p for which the cardinality of E(F_p) is…

In this paper we study elliptic curves which have a number of points whose coordinates are in arithmetic progression. We first motivate this diophantine problem, prove some results, provide a number of interesting examples and, finally…

The Fermat numbers have many notable properties, including order universality, coprimality, and definition by a recurrence relation. We use arbitrary elliptic curves and rational points of infinite order to generate sequences that are…

We prove that there exist infinitely many elliptic curves over $\mathbb{Q}(i)$ with $j$-invariant $1728$ and rank exactly $2$ which are not obtained by base change from $\mathbb{Q}$. The rank of each such curve is determined via 2-isogeny…

We show that the average number of integral points on elliptic curves, counted modulo the natural involution on a punctured elliptic curve, is bounded from above by $2.1 \times 10^8$. To prove it, we design a descent map, whose prototype…

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…

We give explicit parametric equations for all irreducible plane projective sextic curves which have at most double points and whose total Milnor number is maximal (is equal to 19). In each case we find a parametrization over a number field…

We prove that there exist infinitely many elliptic curves over \Q with given modular invariant, and rank >=2. Furthermore, there exist infinitely many elliptic curves over $\Q$ with invariant equal at 0 (resp. 1728) and rank >=6 (resp.…

Let P(x,y) be a rational polynomial and k in Q be a generic value. If the curve (P(x,y)=k) is irreducible and admits an infinite number of points whose coordinates are integers then there exist algebraic automorphisms that send P(x,y) to…

We give some heuristics for counting elliptic curves with certain properties. In particular, we re-derive the Brumer-McGuinness heuristic for the number of curves with positive/negative discriminant up to $X$, which is an application of…

We consider the structure of rational points on elliptic curves in Weierstrass form. Let x(P)=A_P/B_P^2 denote the $x$-coordinate of the rational point P then we consider when B_P can be a prime power. Using Faltings' Theorem we show that…

We solve the problem of characteristic numbers of elliptic curves in any dimensional projective space The answers are given in the form of effective recursions. Many numerical examples are provided. A C++ program implementing all the…

It is well-known that every elliptic curve over the rationals admits a parametrization by means of modular functions. In this short note, we show that only finitely many elliptic curves over $\mathbf{Q}$ can be parametrized by modular…