It is apparent that the abc conjecture is deeply related to Arakelov theory. In one direction, it is shown in S. Lang, "Introduction to Arakelov Theory", that a certain height inequality in Arakelov theory implies the abc conjecture. I am wondering about the other direction and precise implications. Are there such results?
More importantly, if there exist such results, what are some Diophantine implications? That is, what are the Diophantine implications of abc conjecture, that factor through Arakelov Theory/Arithmetic Geometry?
(I once read that Joseph Oesterle came up with abc conjecture while trying to do computations towards the Taniyama-Shimura conjecture. But that story does not make clear the connection with Arakelov theory.)