We discuss the Mordell-Weil sieve as a general technique for proving results concerning rational points on a given curve. In the special case of curves of genus 2, we describe quite explicitly how the relevant local information can be obtained if one does not want to restrict to mod p information at primes of good reduction. We describe our implementation of the Mordell-Weil sieve algorithm and...

In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses we can find m...

We will give an upper bound of Mordell-Weil rank r for relatively minimal brations of curves of genus g 1 on rational surfaces. Under the assumption that a bration is not locally trivial, we have r 4g+4. Moreover the maximal case (r = 4g + 4) will be studied in detail. We determine the structure of such brations and also the structure of their Mordell-Weil lattices introduced by Shioda.

by the mordell-weil theorem‎, ‎the group of rational points on an elliptic curve over a number field is a finitely generated abelian group‎. ‎there is no known algorithm for finding the rank of this group‎. ‎this paper computes the rank of the family $ e_p:y^2=x^3-3px $ of elliptic curves‎, ‎where p is a prime‎.

In this paper, we consider the average size of the 2-Selmer groups of a class of quadratic twists of each elliptic curve over Q with Q-torsion group Z2 × Z2. We prove the existence of a positive proportion of quadratic twists of such a curve, each of which has rank 0 Mordell-Weil group.

We explain how recent work on 3-descent and 4-descent for elliptic curves over Q can be combined to search for generators of the Mordell-Weil group of large height. As an application we show that every elliptic curve of prime conductor in the SteinWatkins database has rank at least as large as predicted by the conjecture of Birch and Swinnerton-Dyer.

We describe in terms of the j-invariant all elliptic surfaces π : X → C with a section, such that h(X) = rankNS(X) and the Mordell-Weil group of π is finite. We use this to give a complete solution to infinitesimal Torelli for elliptic surfaces over P with a section.