Related papers: Quadratic twists of pairs of elliptic curves

Let $K=\mathbb{Q}(\sqrt{-p})$ be a quadratic field for an odd prime $p$. We show that there exist infinitely many primes $p$ for which no elliptic curve $E/\mathbb{Q}$ has torsion subgroup $\mathbb{Z}/2\mathbb{Z}\times…

Let $K = \mathbb{Q}(\sqrt{-3})$ or $\mathbb{Q}(\sqrt{-1})$ and let $C_n$ denote the cyclic group of order $n$. We study how the torsion part of an elliptic curve over $K$ grows in a quadratic extension of $K$. In the case $E(K)[2] \approx…

Given an elliptic curve $E/\mathbb{Q}$, it is a conjecture of Goldfeld that asymptotically half of its quadratic twists will have rank zero and half will have rank one. Nevertheless, higher rank twists do occur: subject to the parity…

In 2006, Elkies presented an elliptic curve with 28 independent rational points. We prove that subject to GRH, this curve has Mordell-Weil rank equal to 28 and analytic rank at most 28. We prove similar results for a previously unpublished…

Let $E$ be an elliptic curve, defined over a quartic extension $K$ of $\mathbb{Q}$, with $j(E) \in \mathbb{Q}$. In this paper, we classify the possible torsion subgroup structures $E(K)_{\text{tors}}$.

In this paper we prove a relation between the period of an elliptic curve and the period of its real and imaginary quadratic twists. This relation is often misstated in the literature.

Let $K$ be a number field, $\bar{K}$ an algebraic closure of $K$ and $E/K$ an elliptic curve defined over $K$. In this paper, we prove that if $E/K$ has a $K$-rational point $P$ such that $2P\neq O$ and $3P\neq O$, then for each $\sigma\in…

For any number field $K$ and integer $0\leq r \leq 4$, we prove that there are infinitely many elliptic curves over $K$ of rank $r$. Our elliptic curves are obtained by specializing well-chosen nonisotrivial elliptic curves over the…

Let $E$ be an elliptic defined over a number field $K$. Then its Mordell-Weil group $E(K)$ is finitely generated: $E(K)\cong E(K)_{tor}\times\mathbb{Z}^r$. In this paper, we discuss the cyclic torsion subgroup of elliptic curves over cubic…

We study the rank of the modular curve $X_0(49)$ over quadratic extensions. Assuming the Birch and Swinnerton-Dyer Conjecture, we show that the rank over $\mathbb{Q}(\sqrt{d})$ is positive if and only if the number of solutions of two…

We study the problem of the existence of arithmetic progressions of three cubes over quadratic number fields Q(sqrt(D)), where D is a squarefree integer. For this purpose, we give a characterization in terms of Q(sqrt(D))-rational points on…

We establish a congruence formula between $p$-adic logarithms of Heegner points for two elliptic curves with the same mod $p$ Galois representation. As a first application, we use the congruence formula when $p=2$ to explicitly construct…

We present an example of a fibred quadratic polynomial admitting an attracting invariant 2-curve. By an unfolding construction we obtain an example of a fibred quadratic polynomial admitting two attracting invariant curves. This phenomena…

We study the distribution of ranks of elliptic curves in quadratic twist families using Iwasawa-theoretic methods, contributing to the understanding of Goldfeld's conjecture. Given an elliptic curve $ E/\mathbb{Q} $ with good ordinary…

We prove that for every number field $K$, there exist infinitely many elliptic curves $E$ over $K$ with rank exactly equal to 1.

We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We prove that the fraction of twists (of a given elliptic curve over a fixed number field) having…

We show that the unboundedness of the ranks of the quadratic twists of an elliptic curve is equivalent to the divergence of certain infinite series.

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 $E/\mathbb{Q}$ be an elliptic curve with full rational 2-torsion. As d varies over squarefree integers, we study the behaviour of the quadratic twists $E_d$ over a fixed quadratic extension $K/\mathbb{Q}$. We prove that for 100% of…

We determine all modular curves $X_0(N)$ with infinitely many quartic points. To do this, we define a pairing that induces a quadratic form representing all possible degrees of a rational morphism from $X_0(N)$ to a positive rank elliptic…