Vojta introduced a dictionary between value distribution theory? of Nevanlinna and Diophantine approximation theory? of Rothand and suggested that this dictionary should continue to hold in higher dimensions. This leads to qualitative conjectures in arithmetic geometry which cover almost every important conjecture in the field; notably the abc conjecture, Mordell's conjecture, some of Lang's conjecture?s. The perspective given by Arakelov theory is central in Vojta’s conjectures.

Paul Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, 1239, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0072989, ISBN 978-3-540-17551-3, MR883451

See also

David McKinnon, Vojta’s main conjecture for blowup surfaces (pdf)

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