17 citations · 49 across the 11 of their papers we have counts for
13 papers · 1 filter
Uniformity in rational torsion and small points on abelian varieties
Ziyang Gao, Kaiyuan Gu
In this paper, we propose a method to study the {\it Uniform Boundedness Conjecture} and the {\it Lang-Silverman Conjecture} for abelian varieties defined over a global field $…
Degeneracy loci in the universal family of abelian varieties
Ziyang Gao, Philipp Habegger
Recent developments on the uniformity of the number of rational points on curves and subvarieties in a moving abelian variety rely on the geometric concept of the degeneracy locus.…
The Relative Manin-Mumford Conjecture
Ziyang Gao, Philipp Habegger
We prove the Relative Manin-Mumford Conjecture for families of abelian varieties in characteristic 0. We follow the Pila-Zannier method to study special point problems, and we use…
The Uniform Mordell-Lang Conjecture
Ziyang Gao, Tangli Ge, Lars Kühne
The Mordell--Lang conjecture for abelian varieties states that the intersection of an algebraic subvariety with a subgroup of finite rank is contained in a finite union of cose…
Recent developments of the Uniform Mordell-Lang Conjecture
Ziyang Gao
This expository survey is based on my online talk at the ICCM 2020. It aims to sketch key steps of the recent proof of the uniform Mordell-Lang conjecture for curves embedded into…
The Ax-Schanuel conjecture for variations of mixed Hodge structures
Ziyang Gao, Bruno Klingler
We prove in this paper the Ax-Schanuel conjecture for all admissible variations of mixed Hodge structures.
A consequence of the relative Bogomolov conjecture
Vesselin Dimitrov, Ziyang Gao, Philipp Habegger
We propose a formulation of the relative Bogomolov conjecture and show that it gives an affirmative answer to a question of Mazur's concerning the uniformity of the Mordell-Lang co…
Uniformity in Mordell-Lang for curves
Vesselin Dimitrov, Ziyang Gao, Philipp Habegger
Consider a smooth, geometrically irreducible, projective curve of genus defined over a number field of degree . It has at most finitely many rational points by t…
Uniform bound for the number of rational points on a pencil of curves
Vesselin Dimitrov, Ziyang Gao, Philipp Habegger
Consider a one-parameter family of smooth, irreducible, projective curves of genus defined over a number field. Each fiber contains at most finitely many rational points b…
Generic rank of Betti map and unlikely intersections
Ziyang Gao
Let be an abelian scheme over an irreducible variety over of relative dimension . For any simply-connected subset of $S^{\mathrm{an}…
Mixed Ax-Schanuel for the universal abelian varieties and some applications
Ziyang Gao
In this paper we prove the mixed Ax-Schanuel theorem for the universal abelian varieties (more generally any mixed Shimura variety of Kuga type), and give some simple applications.…
Heights in families of abelian varieties and the Geometric Bogomolov Conjecture
Ziyang Gao, Philipp Habegger
On an abelian scheme over a smooth curve over a symmetric relatively ample line bundle defines a fiberwise Néon-Tate height. If the base curve is inside a pr…
A special point problem of André-Pink-Zannier in the universal family of abelian varieties
Ziyang Gao
The André-Pink-Zannier conjecture concerns the intersection of subvarieties and the generalized Hecke orbit of a given point in mixed Shimura varieties. It is part of the Zilber-Pi…