- Are all the curves of genus one representing elements of
the Shafarevich-Tate group of
*E*isomorphic (over**Q**) to curves contained in a (single) abelian surface*A*, which is itself defined over**Q**, which contains*E*as a sub-elliptic curve, and which is in turn contained in the new part of the jacobian of a modular curve*X_0(N)*?

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.

- visual.dvi (90528 bytes) [1998 Oct 8]
- visual.dvi.gz (35756 bytes)
- visual.ps.gz (86720 bytes)

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