Mixed Ax–Schanuel for the universal abelian varieties and some applications

Gao, Ziyang

doi:10.1112/s0010437x20007447

Cited by 21 publications

(29 citation statements)

References 22 publications

(53 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For an abelian scheme X → S of relative dimension g with a highdimensional base S, there are natural generalizations of the above situation by André-Corvaja-Zannier [ACZ] and Gao [Gao1]. Namely, by [ACZ,Thm.…”

Section: Arithmetic Casementioning

confidence: 99%

See 1 more Smart Citation

Adelic line bundles over quasi-projective varieties

Yuan¹,

Zhang²

2021

Preprint

View full text Add to dashboard Cite

Section: Arithmetic Casementioning

confidence: 99%

“…2.3.1,Prop. 2.1.1] and [Gao1,Thm. 9.1], a closed subvariety Y of X is nondegenerate and contains a Zariski dense set of torsion points if the following conditions hold:…”

Section: Arithmetic Casementioning

confidence: 99%

Adelic line bundles over quasi-projective varieties

Yuan¹,

Zhang²

2021

Preprint

View full text Add to dashboard Cite

“…Remark 4.5. [MPT19] was extended by Gao [Gao18] to mixed Shimura varieties of Kuga type. Recently the full Ax-Schanuel [K17, Conj.…”

Section: The Ax-schanuel Theorem For Period Mapsmentioning

confidence: 99%

Hodge theory, between algebraicity and transcendence

Klingler¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Step 2 solves this unlikely intersection problem, and the key point is to use [Gao20b,Theorem 1.4] to prove that the union mentioned above is a finite union. In this step the notion of weakly optimal subvarieties introduced by the third-named author and Pila [HP16] is involved.…”

Section: Introductionmentioning

confidence: 99%

Uniformity in Mordell–Lang for curves

2021

Self Cite

View full text Add to dashboard Cite

Consider a smooth, geometrically irreducible, projective curve of genus g ≥ 2 defined over a number field of degree d ≥ 1. It has at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of g, d, and the Mordell-Weil rank of the curve's Jacobian, thereby answering in the affirmative a question of Mazur. In addition we obtain uniform bounds, in g and d, for the number of geometric torsion points of the Jacobian which lie in the image of an Abel-Jacobi map. Both estimates generalize our previous work for 1-parameter families. Our proof uses Vojta's approach to the Mordell Conjecture, and the key new ingredient is the generalization of a height inequality due to the second-and third-named authors. Contents 1. Introduction 1 2. Betti map and Betti form 7 3. Setup and notation for the height inequality 13 4. Intersection theory and height inequality on the total space 16 5. Proof of the height inequality Theorem 1.6 22 6. Preparation for counting points 23 7. Néron-Tate distance between points on curves 29 8. Proof of Theorems 1.1, 1.2, and 1.4 31 Appendix A. The Silverman-Tate Theorem revisited 36 Appendix B. Full version of Theorem 1.6 40 References 47

show abstract

Mixed Ax–Schanuel for the universal abelian varieties and some applications

Abstract: 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. In particular, we present an application for studying the generic rank of the Betti map.

Cited by 21 publications

References 22 publications

Adelic line bundles over quasi-projective varieties

Adelic line bundles over quasi-projective varieties

Hodge theory, between algebraicity and transcendence

Uniformity in Mordell–Lang for curves

Contact Info

Product

Resources

About