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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.)

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On the way, can anyone tell the present status of Yuhan Zha's arxiv paper on this conjecture. arxiv.org/pdf/0901.3042v2 or arxiv.org/abs/0901.3042 –  Unknown Jul 25 '10 at 16:31
    
@Solomon: I was aware of that paper; but steered away from speculating about the correctness. –  Anweshi Jul 25 '10 at 16:33
    
It was a hot topic last year, let's see what others have to say. –  Unknown Jul 25 '10 at 16:42
    
@Solomon: It is probably better not to get into discussions on that paper until some expert says that it is correct. Always papers appear claiming to prove difficult results. –  Anweshi Jul 25 '10 at 17:16
    
(tag number-theory replaced with MO standard nt.number-theory.) Moderators, please remove number-theory from the list of tags. Eschew bifurcation, espouse unification! –  T.. Jul 25 '10 at 19:56

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up vote 4 down vote accepted

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

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So are you implying that Mordell is essentially the only external application of Arakelov theory? –  Anweshi Jul 26 '10 at 16:18
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Certainly not. There are many external applications. However, I think it is true at this time that for every Diophantine (ie., in the number field case) consequence of theorems or conjectures in Arakelov/arithmetic geometry that also is a consequence of ABC, the implication from ABC never "factors through" the hard geometry, it bypasses the geometry formalism. It is possible that will change in the future, but in the present state of knowledge, the answer to your question about "factoring through" is that either no example is known or there is some big news that I missed. –  T.. Jul 27 '10 at 18:56
    
Ah, I see. Thanks. –  Anweshi Jul 28 '10 at 14:00

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