Related papers: Cyclicity of Elliptic Curves Modulo Primes in arit…

For a fixed number field and an elliptic curve defined and semi-stable over this number field, we consider the set of prime numbers p such that the Galois representation attached to the p-torsion points of the elliptic curve is reducible.…

Let $E$ be an elliptic curve over a number field $K$. If for almost all primes of $K$, the reduction of $E$ modulo that prime has rational cyclic isogeny of fixed degree, we can ask if this forces $E$ to have a cyclic isogeny of that degree…

We introduce and study the concept of cyclicity degree of a finite group $G$. This quantity measures the probability of a random subgroup of $G$ to be cyclic. Explicit formulas are obtained for some particular classes of finite groups. An…

Let $p\ge5$ be a prime and $T$ a Kodaira type of the special fiber of an elliptic curve. We estimate the number of elliptic curves over $\mathbb Q$ up to height $X$ with Kodaira type $T$ at $p$. This enables us find the proportion of…

The cyclicity and Koblitz conjectures ask about the distribution of primes of cyclic and prime-order reduction, respectively, for elliptic curves over $\mathbb{Q}$. In 1976, Serre gave a conditional proof of the cyclicity conjecture, but…

This is the first in a series of papers in which we study the n-Selmer group of an elliptic curve, with the aim of representing its elements as genus one normal curves of degree n. The methods we describe are practical in the case n=3 for…

We count by height the number of elliptic curves over the rationals, both up to isomorphism over the rationals and over an algebraic closure thereof, that admit a cyclic isogeny of degree $7$.

We introduce the concept of the modularity of an abelian variety defined over the rational number field extending the modularity of an elliptic curve. We discuss the modularity of an abelian variety over the rational number field. We…

We consider the problem of checking whether an elliptic curve defined over a given number field has complex multiplication. We study two polynomial time algorithms for this problem, one randomized and the other deterministic. The randomized…

We study the collection of group structures that can be realized as a group of rational points on an elliptic curve over a finite field (such groups are well known to be of rank at most two). We also study various subsets of this collection…

The complexity of the elliptic curve method of factorization (ECM) is proven under the celebrated conjecture of existence of smooth numbers in short intervals. In this work we tackle a different version of ECM which is actually much more…

Let $p$ be an odd prime number, $E$ an elliptic curve defined over a number field. Suppose that $E$ has good reduction at any prime lying above $p$, and has supersingular reduction at some prime lying above $p$. In this paper, we construct…

We provide a framework for using elliptic curves with complex multiplication to determine the primality or compositeness of integers that lie in special sequences, in deterministic quasi-quadratic time. We use this to find large primes,…

We observe that there are elliptic curves over number fields all of whose quadratic twists must have positive rank, assuming the Birch-Swinnerton-Dyer conjecture. We give a classification of such curves in terms of their local behaviour,…

We consider the structure of rational points on elliptic curves in Weierstrass form. Let x(P)=A_P/B_P^2 denote the $x$-coordinate of the rational point P then we consider when B_P can be a prime power. Using Faltings' Theorem we show that…

Let $K_i$ be a number field for all $i \in \mathbb{Z}_{> 0}$ and let $\mathcal{E}$ be a family of elliptic curves containing infinitely many members defined over $K_i$ for all $i$. Fix a rational prime $p$. We give sufficient conditions for…

We give an elementary introduction, through illustrative examples but without proofs, to one of the basic consequences of the Langlands programme, namely the law governing the primes modulo which a given irreducible integral polynomial…

Let $E$ be an elliptic curve without complex multiplication defined over the rationals. The purpose of this article is to define a positive integer $A(E)$, that we call the {\it Serre's constant associated to $E$}, that gives necessary…

We give an elementary proof of the fact that any elliptic curve $E$ over an algebraically closed non-archimedean field $K$ with residue characteristic $\neq{2,3}$ and with $v(j(E))<0$ admits a tropicalization that contains a cycle of length…

Let $E$ be an elliptic curve over $\mathbb{Q}$. Then, we show that the average analytic rank of $E$ over cyclic extensions of degree $l$ over $\mathbb{Q}$ with $l$ a prime not equal to $2$, is at most $2+r_{\mathbb{Q}}(E)$, where…