Related papers: Incidence Obstructions for Power Products in Ellip…

We explicitly describe the splitting of odd integral primes in the radical extension $\mathbb{Q}(\sqrt[n]{a})$, where $x^n-a$ is an irreducible polynomial in $\mathbb{Z}[x]$. Our motivation is to classify common index divisors, the primes…

For an elliptic curve $E$ defined over a number field $K$, the heuristic density of the set of primes of $K$ for which $E$ has cyclic reduction is given by an inclusion-exclusion sum $\delta_{E/K}$ involving the degrees of the $m$-division…

Given an elliptic curve $E$ over a finite field $\F_q$ of $q$ elements, we say that an odd prime $\ell \nmid q$ is an Elkies prime for $E$ if $t_E^2 - 4q$ is a quadratic residue modulo $\ell$, where $t_E = q+1 - #E(\F_q)$ and $#E(\F_q)$ is…

Analogously to primes in arithmetic progressions to large moduli, we can study primes that are totally split in extensions of $\mathbb{Q}$ of high degree. Motivated by a question of Kowalski we focus on the extensions $\mathbb{Q}(E[d])$…

It is shown that there are finitely many perfect powers in an elliptic divisibility sequence whose first term is divisible by 2 or 3. For Mordell curves the same conclusion is shown to hold if the first term is greater than 1. Examples of…

In 2009, W. D. Banks and I. E. Shparlinski studied the average densities of primes $p \leq x$ for which the reductions of elliptic curves of small height modulo $p$ satisfy certain arithmetic properties, namely cyclicity and divisibility of…

In the present paper, we study various Erd\H{o}s type geometric problems in the setting of the integers modulo $q$, where $q=p^l$ is an odd prime power. More precisely, we prove certain results about the distribution of triangles and…

For a pair $(E,P)$ of an elliptic curve $E/\mathbb{Q}$ and a nontorsion point $P\in E(\mathbb{Q})$, the sequence of \emph{elliptic Fermat numbers} is defined by taking quotients of terms in the corresponding elliptic divisibility sequence…

Erd\H{o}s and Hall defined a pair $(m, n)$ of positive integers to be interlocking, if between any pair of consecutive divisors (both larger than $1$) of $n$ (resp. $m$) there is a divisor of $m$ (resp. $n$). A positive integer is said to…

We define three hard problems in the theory of elliptic divisibility sequences (EDS Association, EDS Residue and EDS Discrete Log), each of which is solvable in sub-exponential time if and only if the elliptic curve discrete logarithm…

Given an elliptic curve $E$ defined over the rational numbers and a prime $p$ at which $E$ has good reduction, we consider the Galois deformation ring parametrizing lifts of the residual representation on the $p$-torsion group $E[p]$. For a…

Let S be a torsion section of an elliptic surface with only I_n fibers. This article addresses the question: which components of singular fibers can S pass through? We give necessary criteria for the "component numbers", and show an…

Let $E$ be an elliptic curve over $\Q$ without complex multiplication, and which is not isogenous to a curve with non-trivial rational torsion. For each prime $p$ of good reduction, let $|E(\F_p)|$ be the order of the group of points of the…

We introduce the notion of $\epsilon$-irreducibility for arithmetic cycles meaning that the degree of its analytic part is small compared to the degree of its irreducible classical part. We will show that for every $\epsilon>0$ any…

We call a pair of distinct prime powers $(q_1,q_2) = (p_1^{a_1},p_2^{a_2})$ a Hasse pair if $|\sqrt{q_1}-\sqrt{q_2}| \leq 1$. For such pairs, we study the relation between the set $\mathcal{E}_1$ of isomorphism classes of elliptic curves…

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…

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…

Given a subset $S$ of the non-identity elements of the dihedral group of order $2m$, is it possible to order the elements of $S$ so that the partial products are distinct? This is equivalent to the sequenceability of the group when $|S| =…

Let $E$ be an elliptic curve defined over a number field $K$. We say that a prime number $p$ is exceptional for $(E,K)$ if $E$ admits a $p$-isogeny defined over $K$. The so-called exceptional set of all such prime numbers is finite if and…

A conjecture of Erd\H{o}s states that, for any large prime $q$, every reduced residue class $\pmod q$ can be represented as a product $p_1p_2$ of two primes $p_1,p_2\leq q$. We establish a ternary version of this conjecture, showing that,…