The Uniform Mordell–Lang Conjecture

Ziyang Gao; Tangli Ge; Lars Kühne

doi:10.5802/pmihes.26

¹Department of Mathematics UCLA Los Angeles, CA 90095, USA
²Department of Mathematics Princeton University Princeton, NJ 08544, USA
³School of Mathematics and Statistics University College Dublin Dublin 4, Ireland

Publications Mathématiques de l'IHÉS, Online first, pp. 189-235

Abstract

The Mordell–Lang conjecture for abelian varieties states that the intersection of an algebraic subvariety $X$ with a subgroup of finite rank is contained in a finite union of cosets contained in $X$. In this article, we prove a uniform version of this conjecture, meaning that the number of cosets necessary depend only on the dimension of the ambient variety and the degree (with respect to some polarization) of the subvariety $X$. To achieve this, we prove a general gap principle on algebraic points that extends the new gap principle for curves embedded into their Jacobians, previously obtained by Dimitrov–Gao–Habegger and Kühne. Our new gap principle also implies the full uniform Bogomolov conjecture in abelian varieties.

Received: 2023-07-29
Revised: 2026-03-11
Accepted: 2026-03-25
Online First: 2026-04-02

DOI: 10.5802/pmihes.26

Classification: 11G10, 14G05, 14G25, 14K12

Author's affiliations:

Ziyang Gao ¹ ; Tangli Ge ² ; Lars Kühne ³

¹ Department of Mathematics UCLA Los Angeles, CA 90095, USA
² Department of Mathematics Princeton University Princeton, NJ 08544, USA
³ School of Mathematics and Statistics University College Dublin Dublin 4, Ireland

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     author = {Ziyang Gao and Tangli Ge and Lars K\"uhne},
     title = {The {Uniform} {Mordell{\textendash}Lang} {Conjecture}},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     year = {2026},
     publisher = {IHES},
     doi = {10.5802/pmihes.26},
     language = {en},
     note = {Online first},
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Ziyang Gao; Tangli Ge; Lars Kühne. The Uniform Mordell–Lang Conjecture. Publications Mathématiques de l'IHÉS, Online first, pp. 189-235

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