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 |
Keywords:
ABC conjecture; the error term in the ABC conjecture; radicalized Vojta height inequality; Diophantine approximation; Roth’s theorem; type of an algebraic number; Mordell’s conjecture; effective MordellCitations:
Zbl 1083.11042References:
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