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ABC implies the radicalized Vojta height inequality for curves. (English) Zbl 1197.11087

Summary: The truncated or radicalized counting function of a meromorphic function \(f : \mathbb C \to \mathbb P ^{1}(\mathbb C)\) counts the number of times that \(f\) takes a value \(a\), but without multiplicity. By analogy, one also defines this function for numbers. In this sequel to the author’s paper in J. Number Theory 95, 289-302 (2002; Zbl 1083.11042), we prove the radicalized version of Vojta’s height inequality, using the ABC conjecture. We explain the connection with a conjecture of Serge Lang about the different error terms associated with Vojta’s height inequality and with the radicalized Vojta height inequality.

MSC:

11J97 Number-theoretic analogues of methods in Nevanlinna theory (work of Vojta et al.)
11D45 Counting solutions of Diophantine equations
11D75 Diophantine inequalities
14G40 Arithmetic varieties and schemes; Arakelov theory; heights

Citations:

Zbl 1083.11042
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References:

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