Height inequalities and canonical class inequalities

This lecture is concerned with apriori bounds on the size of solutions to Diophantine equations. We will present a rather random collection of results with the intention to provide a flavor of the subject, with only a glimpse of something like a unifying principle. In the next lecture, we will make a better attempt to give coherence to our exposition, by describing the relation to canonical class inequalities from the theory of algebraic surface. The beginning of the lecture is intended to be entirely popular, while the technical language will grow denser towards the end. It might be best to begin with an example from ordinary number theory which is likely to be more familiar than the arithmetic of function-fields, although it is with the latter that we will be most concerned, and for which there are definite results of a general nature. Consider the equation