Visualizing elements of order three in the Shafarevich-Tate group

1. Introduction. If we wish to write the equations of curves of genus 1 that give elements of the Shafarevich-Tate group of an elliptic curve over a number field K, a choice of ways is open to us. For example, if the element in question is of order 3 the curve of genus 1 corresponding to it occurs as a smooth plane cubic curve over K. In a recent article [C-M] we raised the question of when one can find the curves of genus 1 corresponding to at least some of the elements of Shafarevich-Tate groups as curves in abelian surfaces over K. Adam Logan, using data and programs due to Cremona, studied this question numerically for semistable optimal elliptic curves E over Q of square-free conductor N less than about 3000, and for the odd part of their Shafarevich-Tate group. By an " optimal " elliptic curve (or in older terminology, a " strong Weil " curve) we mean that there is a modular parametrization φ : J 0 (N) → E where N = the conductor of E, J 0 (N) = the jacobian of the modular curve X 0 (N), and such that the kernel of φ is an abelian variety. Any modular elliptic curve is isogeneous, over Q, to a (unique) optimal elliptic curve, and any optimal elliptic curve of conductor N is isomorphic, over Q, to an elliptic curve in J 0 (N). Logan studied the elements of the Shafarevich-Tate group of such optimal elliptic curves, and sought, in effect, to realize the corresponding curves of genus 1 in question as subcurves defined over Q within abelian surfaces contained in the new part of J 0 (N). If E is an optimal elliptic curve for which we can successfully do this for each of the elements of the Shafarevich-Tate group of E, let us say that we have visualized the Shafarevich-Tate group of E in abelian surfaces in the new part of the modular jacobian.