Related papers: On the Sum-Product Problem on Elliptic Curves

For an elliptic curve $E/\mathbb{Q}$ we show that there are infinitely many cyclic sextic extensions $K/\mathbb{Q}$ such that the Mordell-Weil group $E(K)$ has rank greater than the subgroup of $E(K)$ generated by all the $E(F)$ for the…

In the early 2000s, Ramakrishna asked the question: For the elliptic curve $$ E: y^2 = x^3 - x, $$ what is the density of primes $p$ for which the Fourier coefficient $a_p(E)$ is a cube modulo $p$? As a generalization of this question,…

Let K be a finite field. We know that a half of elements of K* is a square. So it is natural to ask how many of them appear as x-coordinate of points on an elliptic curve over K. We consider a specific class of elliptic curves over finite…

Let $\mathcal{X}$ be a projective irreducible nonsingular algebraic curve defined over a finite field $\mathbb{F}_q$. This paper presents a variation of the St\"orh-Voloch theory and sets new bounds to the number of…

In this paper we study the problem of how to determine all elliptic curves defined over an arbitrary number field $K$ with good reduction outside a given finite set of primes $S$ of $K$ by solving $S$-unit equations. We give examples of…

In this paper we prove new bounds for sums of convex or concave functions. Specifically, we prove that for all $A,B \subseteq \mathbb R$ finite sets, and for all $f,g$ convex or concave functions, we have $$|A + B|^{38}|f(A) + g(B)|^{38}…

Let $E/\mathbb{Q}$ be an elliptic curve. For a prime $p$ of good reduction, let $r(E,p)$ be the smallest non-negative integer that gives the $x$-coordinate of a point of maximal order in the group $E(\mathbb{F}_p)$. We prove unconditionally…

We show that if $E/\mathbb{Q}$ is an elliptic curve without complex multiplication and for which there is a prime $q$ such that the image of $\bar{\rho}_{E,q}$ is contained in the normaliser of a split Cartan subgroup of…

Let $\mathbb{F}_q$ denote a finite field of order $q$. A rational function $r(x)\in \mathbb{Q}(x)$ is said to be arithmetically exceptional if it induces a permutation on $\mathbb{P}^1(\mathbb{F}_p)$ for infinitely many primes $p$. Based on…

Assuming Lang's conjectured lower bound on the heights of non-torsion points on an elliptic curve, we show that there exists an absolute constant C such that for any elliptic curve E/Q and non-torsion point P in E(Q), there is at most one…

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…

Let $\mathbb F_{q^2}$ be the finite field with $q^2$ elements. We provide a simple and effective method, using reciprocal polynomials, for the construction of algebraic curves over $\mathbb F_{q^2}$ with many rational points. The curves…

In this article, for the finite field $\mathbb{F}_q$, we show that the $\mathbb{F}_q$-algebra $\mathbb{F}_q[x]/\langle f(x) \rangle$ is isomorphic to the product ring $\mathbb{F}_q^{\deg f(x)}$ if and only if $f(x)$ splits over…

Let $E$ be the elliptic curve $y^2=x(x+1)(x+t)$ over the field $\Fp(t)$ where $p$ is an odd prime. We study the arithmetic of $E$ over extensions $\Fq(t^{1/d})$ where $q$ is a power of $p$ and $d$ is an integer prime to $p$. The rank of $E$…

We review the main conjecture for an elliptic curve on $\Q$ having good supersingular reduction at $p$ and give some consequences of it. Then we define the notion of $\lambda$-invariant and of $\mu$- invariant in this situation,…

Let $p$ and $q$ be two distinct odd primes, $p<q$ and $E_{p,q}:y^2=x^3-pqx$ be an elliptic curve. Fix a line $L_{a.b}:y=\frac{a}{b}x$ where $a\in \mathbb{Z},b\in \mathbb{N}$ and $(a,b)=1$. We study sufficient conditions that $p$ and $q$…

Given a large finite point set, $P\subset \mathbb R^2$, we obtain upper bounds on the number of triples of points that determine a given pair of dot products. That is, for any pair of positive real numbers, $(\alpha, \beta)$, we bound the…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. In this article, we classify all groups that can arise as $E(\mathbb{Q}(\zeta_p))_{\text{tors}}$ up to isomorphism for any prime $p$. When $p - 1$ is not divisible by small integers…

In this paper, we analyze the theta series associated to the quadratic form $Q(\mathbf{x}) := x_1^2 + x_2^2 + x_3^2 + x_4^2$ with congruence conditions on $x_i$ modulo $2, 3, 4$, and $6$. By employing special operators on modular,…

By considering a one-parameter family of elliptic curves defined over $\mathbb{Q}$, we might ask ourselves if there is any bias in the distribution (or parity) of the root numbers at each specialization. From the work of Helfgott, we know…