Generalized Greatest Common Divisors, Divisibility Sequences, and Vojta's Conjecture for Blowups

Details Generalized Greatest Common Divisors, Divisibility Sequences, and Vojta's Conjecture for Blowups Generalized Greatest Common Divisors, Divisibility Sequences, and Vojta's Conjecture for Blowups

2004-07-26

arxiv.org/abs/math/0407415v3

Monatsh. Math. 145 (2005), no. 4, 333--350. (MR2162351)

Mathematics

Number Theory

20 pages; minor corrections (thanks to D. Abramovich, P. Corvaja, U. Zannier)

Scientific paper

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.

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