Topic
Vojta's conjecture
About: Vojta's conjecture is a research topic. Over the lifetime, 37 publications have been published within this topic receiving 1845 citations.
Papers
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29 Aug 1983
TL;DR: In this article, the authors present a review of Lang's Diophantine Geometry, by L. J. Mordell and S. Lang, as well as a discussion of the relation between absolute values and proper sets of absolute values.
Abstract: 1 Absolute Values.- 2 Proper Sets of Absolute Values. Divisors and Units.- 3 Heights.- 4 Geometric Properties of Heights.- 5 Heights on Abelian Varieties.- 6 The Mordell-Weil Theorem.- 7 The Thue-Siegel-Roth Theorem.- 8 Siegel's Theorem and Integral Points.- 9 Hilbert's Irreducibility Theorem.- 10 Weil Functions and Neron Divisors.- 11 Neron Functions on Abelian Varieties.- 12 Algebraic Families of Neron Functions.- 13 Neron Functions Over the Complex Numbers.- Review of S. Lang's Diophantine Geometry, by L. J. Mordell.- Review of L. J. Mordell's Diophantine Equations, by S. Lang.
1,005 citations
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01 Apr 1987
TL;DR: In this article, the main conjecture and the ramification term of the conjecture are discussed. But they do not consider the relation between Nevanlinna theory and hyperplanes, unlike the main conjectures of this paper.
Abstract: Heights and integral points.- Diophantine approximations.- A correspondence with Nevanlinna theory.- Consequences of the main conjecture.- The ramification term.- Approximation to hyperplanes.
444 citations
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TL;DR: In this paper, a new formulation of the bounds in terms of height functions and algebraic subgroups of Gm2 has been proposed for multiplicatively independent S-units u,v∈Z.
Abstract: Let a,b be given, multiplicatively independent positive integers and let e>0. In a recent paper jointly with Y. Bugeaud we proved the upper bound exp(en) for g.c.d.(an−1, bn−1); shortly afterwards we generalized this to the estimate g.c.d.(u−1,v−1)
73 citations
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TL;DR: In this paper, the authors apply Vojta?s conjecture to blowups and deduce a number of deep statements regarding (generalized) greatest common divisors on varieties, in particular on projective space and on abelian varieties.
Abstract: We apply Vojta?s conjecture to blowups and deduce a number of deep statements regarding (generalized) greatest common divisors on varieties, in particular on projective space and on abelian varieties. Special cases of these statements generalize earlier results and conjectures. We also discuss the relationship between generalized greatest common divisors and the divisibility sequences attached to algebraic groups, and we apply Vojta?s conjecture to obtain a strong bound on the divisibility sequences attached to abelian varieties of dimension at least two.
49 citations
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TL;DR: In this article, the authors apply Vojta's conjecture to blowups and deduce a number of deep statements regarding (generalized) greatest common divisors on varieties, in particular on projective space and on abelian varieties.
Abstract: We apply Vojta's conjecture to blowups and deduce a number of deep statements regarding (generalized) greatest common divisors on varieties, in particular on projective space and on abelian varieties. Special cases of these statements generalize earlier results and conjectures. We also discuss the relationship between generalized greatest common divisors and the divisibility sequences attached to algebraic groups, and we apply Vojta's conjecture to obtain a strong bound on the divisibility sequences attached to abelian varieties of dimension at least two.
38 citations