Barry Mazur

Let E be an elliptic curve over Q of conductor N contained, as sub-abelian variety, in J_0(N), the jacobian of the modular curve X_0(N). Almost two years ago I gave a lecture in this seminar about trying to "visualize" some of the elements of the Shafarevich-Tate group of such an elliptic curve E, as curves of genus 1 (defined over Q) contained in the algebraic variety J_0(N). The surprise of computations for squarefree N and odd Sha made by Adam Logan, using data of Cremona, is that so many elements of Sha can be seen this way, and more specifically can be found in abelian surfaces A contained in J_0(N), where A is isogenous to a product E x F (with F another elliptic curve which satisfies a congruence with E, and which has Mordell-Weil rank > 1). As Logan showed, 2849 is the first squarefree conductor N for which there is an elliptic curve in J_0(N) whose (odd) Sha cannot be visualized in this way. Subsequently, Cremona and I wrote a paper extending the computations to include all conductors, squarefree or not, up to 5500, and studying even Sha as well. For this extended range, one still has very few cases where Sha is not visualizable in the way I described. This deserves explanation! In this lecture I will prove that over any number field K any element of order three in any elliptic curve E over K can be represented by a curve of genus 1 (defined over K) in an abelian surface A which is isogenous to E x F (all, of course, over K). This is hardly sufficient to explain the surprising data that has been gathered because the result to be proved gives NO information about the conductor of A, but, at least, it is something...