arXiv subjects

Tangli Ge

Publications and source records attributed to Tangli Ge.

4 recordsLinked to original sources

arXiv2024

Intersecting subvarieties of abelian schemes with group subschemes I

In this paper, we establish the following family version of Habegger's bounded height theorem on abelian varieties: a locally closed subvariety of an abelian scheme with Gao's $t^{\mathrm{th}}$ degeneracy locus removed, intersected with all flat group subschemes of relative dimension at most $t$, gives a set of bounded total height. Our main tools include the Ax--Schanuel theorem, and intersection theory of adelic line bundles as developed by Yuan--Zhang. As two applications, we generalize Silverman's specialization theorem to a higher dimensional base, and establish a bounded height result towards Zhang's ICM Conjecture.

math.NT↗

arXiv2022

Uniform Mordell-Lang Plus Bogomolov

In this paper, we prove a uniform version of Poonen's "Mordell-Lang Plus Bogomolov" theorem for abelian varieties. We mainly generalize R\'emond's work on large points to allow an extra $\epsilon$-neighborhood. The part on small points follows from an earlier paper, joint with Gao and K\"uhne.

math.NT↗

arXiv2021

The Uniform Mordell-Lang Conjecture

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 that the number of cosets necessary does not depend on the ambient abelian variety. To achieve this, we prove a general gap principle on algebraic points that extends the gap principle for curves embedded into their Jacobians, previously obtained by Dimitrov--Gao--Habegger and K\"{u}hne. Our new gap principle also implies the full uniform Bogomolov conjecture in abelian varieties.

math.NT↗

arXiv2021

Uniformity of quadratic points

In this paper, we extend a uniformity result of Dimitrov-Gao-Habegger to dimension two and use it to get a uniform bound on the set of all quadratic points for non-hyperelliptic non-bielliptic curves in terms of the Mordell-Weil rank.

math.AG↗