Visualizing elements in the Shafarevich-Tate group, by J. E. Cremona and B. Mazur

In this paper we review a number of different ways of ``visualizing'' the elements of the Shafarevich-Tate group of an elliptic curve E over a number field K. We are specifically interested in cases where the elliptic curves are defined over Q, and are subabelian varieties of the new part of the jacobian of a modular curve (specifically, of X_0(N) where N is the conductor of the elliptic curve). Conjecturally any elliptic curve over Q is isogenous to one of these. For a given such E with nontrivial Shafarevich-Tate group, we pose the following question:

At first view, one might imagine that there are few E with nontrivial Shafarevich-Tate group for which this question has an affirmative answer. Very likely, once the order of the Shafarevich-Tate group is large enough, the question will have a negative answer.We present a substantial amount of data covering all modular elliptic curves E, as above with conductors up to 5500 (and with no rational point of order 2). We show that for the vast majority of these cases the above question has an affirmative answer. We are puzzled by this and wonder whether there is some conceptual reason for it.

J. E. Cremona and B. Mazur <, mazur@MATH.HARVARD.EDU>