Related papers: Semi-Magic Squares and Elliptic Curves

We show there exist polynomial bounds on torsion of elliptic curves which come from a fixed geometric isogeny class. More precisely, for an elliptic curve $E_0$ defined over a number field $F_0$, for each $\epsilon>0$ there exist constants…

We classify the possible torsion structures of rational elliptic curves over quintic number fields. In addition, let E be an elliptic curve defined over Q and let G = E(Q)_tors be the associated torsion subgroup. We study, for a given G,…

A new characterization of rational torsion subgroups of elliptic curves is found, for points of order greater than 4, through the existence of solution for systems of Thue equations.

Let $a,b,c$ be distinct positive integers. Set $M=a+b+c$ and $N=abc$. We give an explicit description of the Mordell-Weil group of the elliptic curve $\displaystyle E_{(M,N)}:y^2-Mxy-Ny=x^3$ over $\Q$. In particular we determine the torsion…

Let $E$ be an elliptic curve over $\mathbb{Q}$ which has multiplicative reduction at a fixed prime $p$. For each positive integer $n$ we put $K_n:=\mathbb{Q}(E[p^n])$. The aim of this paper is to extend the author's previous our results…

We prove the existence of infinitely many solutions to an elliptic problem by borrowing the techniques from algebraic topology. The solution(s) thus obtained will also be proved to be bounded.

We prove results towards classifying the possible torsion subgroups of elliptic curves over quadratic fields $\mathbb{Q}(\sqrt{d})$, where $0<d<100$ is a square-free integer, and obtain a complete classification for 49 out of 60 such…

Consider an elliptic curve $E$ over a number field $K$. Suppose that $E$ has supersingular reduction at some prime $\mathfrak{p}$ of $K$ lying above the rational prime $p$. We completely classify the valuations of the $p^n$-torsion points…

Extending and unifying a number of well-known conjectures and open questions, we conjecture that locally elliptic (that is, every element has a bounded orbit) actions by automorphisms of finitely generated groups on finite dimensional…

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…

The V_4-lines for each linearly normal space elliptic curve form the edges of a tetrahedron, however in case the elliptic curve has j=12^3, there exist Z_4-lines in addition. We show the arrangement of V_4 and Z_4-lines explicitly for the…

For each natural number $n$ we determine, both asymptotically and exactly, the maximum number of edges an induced subgraph of order $n$ of the $d$-dimension a grid graph ${\ints}^d$ can have. The asymptotic bound is obtained by using a…

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…

Suppose that $E$ is an elliptic curve defined over $\mathbb{Q}$ without complex multiplication and with conductor $N$. For each positive integer $m$, the action of the absolute Galois group…

We establish asymptotic lower bounds for the number of elliptic curves over $\mathbb{Q}$ with prescribed entanglement of division fields, ordered by naive height. Such elliptic curves are obtained as $1$-parameter families arising from…

As a subproduct of the Schoof-Elkies-Atkin algorithm to count points on elliptic curves defined over finite fields of characteristic p, there exists an algorithm that computes, for l an Elkies prime, l-torsion points in an extension of…

The Torsion Anomalous Conjecture states that an irreducible variety $V$ embedded in a semi-abelian variety contains only finitely many maximal $V$-torsion anomalous varieties. In this paper we consider an irreducible variety embedded in a…

Let $E$ be an elliptic curve over a number field $K$, $c_v$ the Tamagawa number of $E$ at $v$, and let $c_E=\prod_{v}c_v$. Lorenzini proved that $v_{13}(c_E)$ is postive for all elliptic curves over quadratic fields with a point of order…

Let $\ell$ be an odd prime and $d$ a positive integer. We determine when there exists a degree-$d$ number field $K$ and an elliptic curve $E/K$ with $j(E)\in\mathbb{Q}\setminus\{0,1728\}$ for which $E(K)_{\mathrm{tors}}$ contains a point of…

For small odd primes $p$, we prove that most of the rational points on the modular curve $X_0(p)/w_p$ parametrize pairs of elliptic curves having infinitely many supersingular primes. This result extends the class of elliptic curves for…