On a Uniform Bound for the Number of Exceptional Linear Subvarieties in the Dynamical Mordell–lang Conjecture

Let φ : P → P be a morphism of degree d ≥ 2 defined over C. The dynamical Mordell–Lang conjecture says that the intersection of an orbit Oφ(P ) and a subvariety X ⊂ P is usually finite. We consider the number of linear subvarieties L ⊂ P such that the intersection Oφ(P ) ∩ L is “larger than expected.” When φ is the d-power map and the coordinates of P are multiplicatively independent, we prove that there are only finitely many linear subvarieties that are “super-spanned” by Oφ(P ), and further that the number of such subvarieties is bounded by a function of n, independent of the point P and the degree d. More generally, we show that there exists a finite subset S, whose cardinality is bounded in terms of n, such that any n+1 points in Oφ(P )rS are in linear general position in P. 1. The Dynamical Mordell–Lang Conjecture The classical Mordell conjecture says that a curve C of genus g ≥ 2 defined over a number field K has only finitely many K-rational points. One may view C as embedded in its Jacobian J , and then Mordell’s conjecture may be reformulated as saying that C intersects the finitely generated group J(K) in only finitely many points. Taking this viewpoint, Lang conjectured that if Γ ⊂ A is a finitely generated subgroup of an abelian variety A and if X ⊂ A is a subvariety of A, then X ∩ Γ is contained in a finite union of translates of proper abelian subvarieties of A. The Mordell–Lang conjecture for abelian varieties was proven by Faltings [8, 9], building on ideas pioneered by Vojta [18] in his alternative proof of the original Mordell conjecture. The classical Mordell–Lang may be reformulated in dynamical terms as follows. Let P1, . . . , Pr be generators of Γ, and for each 1 ≤ i ≤ r, let Ti : A → A be the translationby-Pi map, i.e., Ti(Q) = Q + Pi. Further let T be the group of self-maps of A generated by T1, . . . , Tr. Then Γ is simply the complete orbit of 0 by the group of maps T , so the Mordell–Lang conjecture is a statement about the intersection of an orbit and a subvariety. The following is a dynamical analogue of the Mordell–Lang conjecture for self-morphisms of algebraic varieties; see [4, 10]. Conjecture 1.1 (Dynamical Mordell–Lang Conjecture). Let φ : V → V be a self-morphism of an algebraic variety defined over C, let X ⊂ V be a subvariety, and let P ∈ V (C). Then { n ≥ 0 : φ(P ) ∈ X } Date: September 1, 2011. 2010 Mathematics Subject Classification. Primary: 37P15; Secondary: 11D45, 37P05.