Elliptic surfaces and intersections of adelic $\mathbb{R}$-divisors

DeMarco, Laura; Mavraki, Niki Myrto

doi:10.48550/arxiv.2012.14529

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“…When dim S = 1 and X is the image of a section, Conjecture 10.1 is equivalent to S. Zhang's conjecture in his 1998 ICM note [Zha98b, §4] if A η is simple and is proved by DeMarco-Mavraki [DM20, Thm.1.4] if A → S is isogenous to a fiber product of elliptic surfaces. The latter proof was simplified and strengthened by DeMarco-Mavraki in [DM21]: in [DM20] the authors reduced their Theorem 1.4 to the case of torsion points treated by [MZ14], whereas in [DM21] the authors proved this result (among other generalizations [DM21, Thm.1.5]) directly.…”

Section: Denote By πmentioning

confidence: 99%

Recent developments of the Uniform Mordell-Lang Conjecture

Gao¹

2021

Preprint

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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 Jacobians (a question of Mazur). The full version of this conjecture is proved by combining Dimitrov-Gao-Habegger [DGH20b] and Kühne [Küh21a]. We include in this survey a detailed proof on how to combine these two results, which was implicitly done in [DGH20a] but not explicitly written in existing literature. At the end of the survey we state some future aspects.[1] When the number field F is fixed, [CHM97, CHM21] proved more: Assuming the widely open Strong Lang Conjecture, the cardinality #C(F ) is bounded above solely in terms of g except for finitely many F -isomorphic classes of curves C of genus g ≥ 2.

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Section: Denote By πmentioning

confidence: 99%

Recent developments of the Uniform Mordell-Lang Conjecture

Gao¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

Elliptic surfaces and intersections of adelic $\mathbb{R}$-divisors

Cited by 1 publication

References 0 publications

Recent developments of the Uniform Mordell-Lang Conjecture

Recent developments of the Uniform Mordell-Lang Conjecture

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