Wenhan Dai

The Uniform Mordell-Lang Program

Institute for Theoretical Sciences - Westlake Mathematics Seminar 2021

• Time: September 25 to 26, 2021.
• Venue: Hua Jia Shan Resort (花家山莊).

Lecture Notes (in chronological order)

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September 25

• Liang Xiao (PKU) – Basic height theory. (notes, video)
  The goal is to introduce Weil’s height machine, Neron-Tate height of abelian varieties, and the application to the Mordell-Weil theorem.

• Shun Tang (Capital Normal Univ) – Basic Arakelov theory. (notes, video)
  This lecture introduces arithmetic divisors and hermitian line bundles over arithmetic varieties, and defines their top intersection numbers. Define height functions associated to hermitian line bundles. Introduce ampleness, bigness, nefness of hermitian line bundles.

• Tong Zhang (East China Normal Univ) – Basics on the moduli spaces of curves. (notes, video)
  This is a review of the definition and GIT construction of the moduli spaces of smooth curves and stable curves. Introduce the tautological divisors (Hodge bundle, boundary divisors) over the moduli space of stable curves. Introduce Noether’s formula.

• Xinyi Yuan (PKU) – Introduction to the Mordell conjecture. (notes, video)
  This is an overview of the proofs of the Mordell conjecture, with some details on Vojta’s proof (simplified by Faltings and Bombieri).

• Junyi Xie (CNRS) – Introduction to Berkovich spaces. (notes, video)
  This is an outline of the theory of Berkovich analytic spaces. It will also cover metrized line bundles, and Green’s functions over Berkovich spaces.

• Junyi Xie (CNRS) – Introduction to the Bogomolov conjecture. (notes, video)
  This is an introduction to various forms of the Bogomolov conjecture, with some details on some lower bound for curves in terms of the phi-invariant following Zhang, Cinkir, and de Jong.

September 26

• Xinyi Yuan (PKU) – Uniform Mordell-Lang and uniform Bogomolov (I). (notes, video)
  This is an introduction to the recent results on the uniform Mordell-Lang problem and the uniform Bogomolov problem of Dimitrov-Habegger-Gao, Kühne and Yuan. We will focus on Yuan’s approach.

• Xinyi Yuan (PKU) – Uniform Mordell-Lang and uniform Bogomolov (II). (notes, video)
  This is an introduction to the recent results on the uniform Mordell-Lang problem and the uniform Bogomolov problem of Dimitrov-Habegger-Gao, Kühne and Yuan. We will focus on Yuan’s approach.

Caution: The recoding videos (currently available at Bilibili) are not committed to be permanently available.