Related papers: Constants in Titchmarsh divisor problems for ellip…

Given non-CM elliptic curves $E_1$ and $E_2$ over $\mathbb{Q}$, we study the natural density of primes $p$ of good reduction for which the orders of the groups $E_1(\mathbb{F}_p)$ and $E_2(\mathbb{F}_p)$ are coprime. This problem may be…

Let $K$ be a totally real field, and let $S$ be a finite set of non-archimedean places of $K$. It follows from the work of Merel, Momose and David that there is a constant $B_{K,S}$ so that if $E$ is an elliptic curve defined over $K$,…

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…

Given an elliptic curve $E$ and a finite Abelian group $G$, we consider the problem of counting the number of primes $p$ for which the group of points modulo $p$ is isomorphic to $G$. Under a certain conjecture concerning the distribution…

Let $\mathbb{F}_q$ be a finite field of odd characteristic $p$. We exhibit elliptic curves over the rational function field $K = \mathbb{F}_q(t)$ whose Tate-Shafarevich groups are large. More precisely, we consider certain infinite…

In this note we compute a constant $N$ that bounds the number of non--primitive divisors in elliptic divisibility sequences over function fields of any characteristic. We improve a result of Ingram--Mah{\'e}--Silverman--Stange--Streng,…

We study the existence of a Zsigmondy bound for a sequence of divisors associated to points on an elliptic curve over a function field. More precisely, let $k$ be an algebraically closed field, let $\mathcal{C}$ be a nonsingular projective…

Major controversy surrounds the use of Elliptic Curves in finite fields as Random Number Generators. There is little information however concerning the "randomness" of different procedures on Elliptic Curves defined over fields of…

Let $p\ge 5$ be a prime number and $E/\mathbf{Q}$ an elliptic curve with good supersingular reduction at $p$. Under the generalized Heegner hypothesis, we investigate the $p$-primary subgroups of the Tate--Shafarevich groups of $E$ over…

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…

Let $\E$ be an elliptic curve over a finite field $\F_{q}$ of $q$ elements, with $\gcd(q,6)=1$, given by an affine Weierstra\ss\ equation. We also use $x(P)$ to denote the $x$-component of a point $P = (x(P),y(P))\in \E$. We estimate…

We give explicit uniform bounds for several quantities relevant to the study of Galois representations attached to elliptic curves $E/\mathbb Q$. We consider in particular the subgroup of scalars in the image of Galois, the first Galois…

We show that for all epsilon > 0, there is a constant C(epsilon) > 0 such that for all elliptic curves E defined over a number field F with j(E) in Q we have #E(F)[tors] \leq C(epsilon)[F:Q]^{5/2+epsilon}. We pursue further bounds on the…

Amicable pairs for a fixed elliptic curve defined over $\mathbb{Q}$ were first considered by Silverman and Stange where they conjectured an order of magnitude for the function that counts such amicable pairs. This was later refined by Jones…

We prove asymptotic formulas for cyclicity of reductions of elliptic curves over the rationals in a family of curves having specified torsion. These results agree with established conditional results and with average results taken over…

It is well-known that if $E$ is an elliptic curve over the finite field $\mathbb{F}_p$, then $E(\mathbb{F}_p)\simeq\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/mk\mathbb{Z}$ for some positive integers $m, k$. Let $S(M,K)$ denote the set of pairs…

Division polynomials associated to an elliptic curve $E/K$ are polynomials $\phi_n, \psi_n^2$ that arise from the sequence of points $\{nP\}_{n \in \mathbb{N}}$ on this curve. If one wishes to study $\mathbb{Z}$--linear combination of…

Let $E$ be an elliptic curve defined over a number field $F$. In this paper, we study the structure of the $p^\infty$-Selmer group of $E$ over $p$-adic Lie extensions $F_\infty$ of $F$ which are obtained by adjoining to $F$ the $p$-division…

In this article, we construct algebraic equations for a curve C and a map f to an elliptic curve E, with pre-specified branching data. We do this by determining certain relations that the periods of C and E must satisfy and use these…

We present normal forms for elliptic curves over a field of characteristic $2$ analogous to Edwards normal form, and determine bases of addition laws, which provide strikingly simple expressions for the group law. We deduce efficient…