We study generalisations to totally real fields of the methods originating with Wiles and Taylor-Wiles ([32], [31]). In view of the results of Skinner-Wiles [26] on elliptic curves with ordinary reduction, we focus here on the case of supersingular reduction. Combining these, we then obtain some partial results on the modularity problem for semistable elliptic curves, and end by giving some app...

The aim of this paper is to introduce a protocol for multicast distribution of secrets providing perfect forward and backward secrecy. The protocol is based on pairings on elliptic curves and has low cost communications and key storage. It is shown how this protocol improves some alternatives existing for real time communications. Keywords—Information security, multicast communications, ellipti...

We construct infinite families of elliptic curves with given torsion group structures over quartic number fields. In a 2006 paper, the first two authors and Park determined all of the group structures which occur infinitely often as the torsion of elliptic curves over quartic number fields. Our result presents explicit examples of their theoretical result. This paper also presents an efficient ...

An Elliptic Curve can be roughly described as the set of solutions of an equation of the form y2 = x3 + ax+ b over some field (e.g. C,R,Q or some finite field Fpn). The importance of elliptic curves stems from their rich structure: there is a rather simple addition law definable on elliptic curves which makes them into an abelian group. Studying the emerging structure of elliptic curves over va...

Several authors have investigated the problem of finding elliptic curves over Q that contain rational points whose x-coordinates are in arithmetic progression. Traditionally, the elliptic curve has been taken in the form of an elliptic cubic or elliptic quartic. Moody studied this question for elliptic curves in Edwards form, and showed that there are infinitely many such curves upon which ther...

CONTENTS Introduction 1. Elliptic Curves Defined by Simplest Cubic Fields 2. Linear Forms in Elliptic Logarithms 3. Computation of Integral Points 4. Tables of Results 5. General Results about Integral Points on the Elliptic Curves y2 = x3 + mx2 (m+3)x + 1 References Let f(X) be a cubic polynomial defining a simplest cubic field in the sense of Shanks. We study integral points on elliptic curve...

This is an introduction to a probabilistic model for the arithmetic of elliptic curves, a model developed in a series of articles of the author with Bhargava, Kane, Lenstra, Park, Rains, Voight, and Wood. We discuss the theoretical evidence for the model, and we make predictions about elliptic curves based on corresponding theorems proved about the model. In particular, the model suggests that ...

In this paper we construct infinite families of elliptic curves with given torsion group structures over cubic number fields. This result provides explicit examples of the theoretical result recently developed by the first two authors and A. Schweizer; they determined all the group structures which occur infinitely often as the torsion of elliptic curves over cubic number fields. In fact, this ...

Abstract. We prove several results regarding some invariants of elliptic curves on average over the family of all elliptic curves inside a box of sides A and B. As an example, let E be an elliptic curve defined over Q and p be a prime of good reduction for E. Let eE (p) be the exponent of the group of rational points of the reduction modulo p of E over the finite field Fp. Let C be the family o...