In this talk we study a modular variant of this problem. In fact we
look at the intersection of a fixed algebraic curve in the affine
plane with the union of all modular curves Y_0(n). In other words, we
study points on the fixed curve whose coordinates are j-invariants of
isogenous elliptic curves. Motivated by the example in algebraic tori
one can ask if such points have height bounded independently of n.
Unfortunately, this is false in all interesting cases.
We state a conjecture giving a height upper bound in terms of
n which is weak enough to deduce some simple finiteness
results. We also discuss a number of cases where this conjecture can