Consider a fixed algebraic curve contained in an algebraic torus and defined over the field of algebraic numbers. It has been known for some time, that points on the fixed curve with multiplicatively dependent coordinates have uniformly bounded height, unless the curve satisfies a simple geometric property. This boundedness of height can be used to prove finiteness results going into the direction of conjectures of Pink and Zilber.

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 be proven.