The Uniform Mordell-Lang Conjecture

Abstract

In this paper, we prove the Uniform Mordell-Lang Conjecture for subvarieties in abelian varieties. As a byproduct, we prove the Uniform Bogomolov Conjecture for subvarieties in abelian varieties. Contents 1. Introduction 1 2. Preliminary knowledge on abelian varieties 5 3. Basic setup: Hilbert schemes and non-degeneracy 8 4. Application of the height inequality 13 5.

Elliptic Curves with Long Arithmetic Progressions Have Large Rank

“…A first breakthrough was obtained in 2020 by DeMarco, Krieger, and Ye [4] when they proved the conjectured uniform bound in the case E 1 and E 2 are given in Legendre form y 2 = x(x − 1)(x − λ) and π j are the corresponding projections onto the x-coordinate. As noted in [6], the Bogomolov-Fu-Tschinkel conjecture is now completely solved thanks to the recent proof of the Uniform Manin-Mumford conjecture [5,9,7,11]. In this work we prove the following generalization: Theorem 1.3 (Main Theorem for torsion).…”

Ranks of abelian varieties in cyclotomic twist families

“…where c g is a constant depending only g [Küh21]. Building on this, Gao, Ge, and Kühne proved the more general uniform Mordell-Lang conjecture [GGK21] for closed subvarieties of abelian varieties. These results reduce the question of uniform bounds for rational points on a large class of varieties to a question about ranks of abelian varieties.…”

“…Since the results of [GGK21] are ineffective, we cannot say anything explicit about the constant N ε in general. However, one can prove explicit results in this direction by instead combining our work with the Chabauty method.…”

Degeneracy loci in the universal family of abelian varieties

“…(This and a converse claim is contained in [Gao20a, Theorem 1.7].) Zariski closedness of X deg (0) is not logically required in the context of [DGH21,Küh21,GGK21].…”

“…In §5 we emphasize the case t = 0. The zeroth degeneracy locus X deg (0) is of crucial importance in the recent proof of the Uniform Mordell-Lang Conjecture [DGH21, Küh21,GGK21]. The mixed Ax-Schanuel Theorem [Gao20b] for the universal family of abelian varieties links the concept of non-degeneracy, in the sense of [DGH21, Definition 1.5], with the size of X deg (0) in X.…”