On the abc conjecture and diophantine approximation by rational points

Details On the abc conjecture and diophantine approximation by rational points On the abc conjecture and diophantine approximation by rational points

1999-08-06

arxiv.org/abs/math/9908024v2

Mathematics

Number Theory

28 pages, 1 figure Some minor errors fixed; updated references to previous work in the field

Scientific paper

We show that an earlier conjecture of the author, on diophantine approximation of rational points on varieties, implies the ``abc conjecture'' of Masser and Oesterl'e. In fact, a weak form of the former conjecture is sufficient, involving an extra hypothesis that the variety and divisor admit a faithful group action of a certain type. Analogues of this weaker conjecture are proved in the split function field case of characteristic zero, and in the case of holomorphic curves (Nevanlinna theory). The proof of the latter involves a geometric generalization of the classical lemma on the logarithmic derivative, due to McQuillan. This lemma may be of independent interest.

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