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.)
Best Answer
ABC is equivalent to the conjectured height inequality that Lang (or more precisely Vojta, in an appendix to Lang's book, following the ABC appendix he wrote for his own book) uses. This is shown in several papers by van Frankenhuijsen.
http://research.uvu.edu/machiel/papers/abcrvhi.pdf
http://research.uvu.edu/machiel/papers/ABCRothMord.pdf
http://research.uvu.edu/machiel/bibliography.html
So ABC is equivalent to some of Vojta's conjectures in arithmetic geometry.
Also, Elkies' "ABC implies effective Mordell" is a sort of counter-application in that it shows that, given the ABC conjecture, one does not need Arakelov methods to prove the Mordell conjecture.
http://imrn.oxfordjournals.org/cgi/pdf_extract/1991/7/99
Lang wrote an article on Diophantine inequalities related to ABC that outlines some of the relationships between conjectures that were known 20 years ago.
http://www.ams.org/bull/1990-23-01/S0273-0979-1990-15899-9/S0273-0979-1990-15899-9.pdf