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

Abstract

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.

Integral points and Vojta’s conjecture on rational surfaces

“…As we will see, this gives us a gcd inequality for integral points on G 2 m . The result for the case of the first blowup of P 2 was previously observed by Silverman [14] to be equivalent to the highly nontrivial inequality of the gcd of polynomial expressions of S-units, proved by Corvaja and Zannier [3]. To prove the multiple blowup case in general, we again use [3] together with some inductive arguments.…”

“…However, proving Vojta's conjecture on the blowup is still often nontrivial even if the conjecture is known for the base: for blowups of abelian varieties, the only known case is the blowup of E × E, where E is a rank-1 elliptic curve (McKinnon [10]). Moreover, Vojta's conjecture on blowups often implies new arithmetic results, including interesting greatest-common-divisor (gcd) inequalities and properties of elliptic divisibility sequences, as Silverman [14] has explored.…”

Ideals of degree one contribute most of the height

“…See [McKinnon 2003] for a proof, and [Silverman 2005] for a discussion of the implications of Vojta's conjecture in this context. .…”

On the Error Terms and Exceptional Sets in Conjectural Second Main Theorems

“…This theorem is essentially a combination of an observation of Silverman in [13] and a remark of Bugeaud, Corvaja, and Zannier in [4].…”